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eISSN: 2378-315X

Biometrics & Biostatistics International Journal

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Received: January 01, 1970 | Published: ,

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Abstract

Length-biased and Area-biased distributions arise in many forestry applications, as well as other environmental, econometric, and biomedical sampling problems. We examine the Length-biased and Area-biased distributions versions of the Exponentiated Weibull distribution (EW). This study introduced a new distribution based on Length-biased Exponentiated Weibull distribution (LBEW) and Area-biased Exponentiated Weibull distribution (ABEW). Some characteristics of the new distributions were obtained. Plots for the cumulative distribution function, pdf and tables with values of skewness and kurtosis were also provided. Height-Diameter (H-D) data on Bombax and Pines (Pinus caribaea) were used to demonstrate the application of the distributions. Estimation of parameters of EW, LBEW and ABEW distributions were done using the maximum likelihood approach and compared across the distributions using criteria like AIC and Loglikelihood. We therefore proposed that similar to Exponentiated Weibull distribution (EW), a better fitting of Bombax and Pines H-D data are possible by LBEW and ABEW distributions. We hope in numerous fields of theoretical and applied sciences, the findings of this study will be useful for the practitioners.

Keywords: length-biased distribution, area-biased distributions, forestry, bombax, pines, height and diameter

Introduction

Trees contribute to the environment by providing oxygen, improving air quality, climate amelioration, conserving water, preserving soil, and supporting wildlife. During the process of photosynthesis, trees take in carbon dioxide and produce the oxygen we breathe. According to the U.S. Department of Agriculture, "One acre of forest absorbs six tons of carbon dioxide and puts out four tons of oxygen. This is enough to meet the annual needs of 18 people." Trees, shrubs and turf also filter air by removing dust and absorbing other pollutants like carbon monoxide, sulphur dioxide and nitrogen dioxide. After trees intercept unhealthy particles, rain washes them to the ground. Trees can add value to your home, help cool your home and neighborhood break the cold winds to lower your heating costs, and provide food for wildlife.

Height-diameter relationships are used to estimate the heights of trees measured for their diameter at breast height (DBH). Such relationship describes the correlation between height and diameter of the trees in a stand on a given date and can be represented by a linear or non-linear statistical model. In forest inventory designs diameter at breast height is measured for all trees within sample plots, while height is measured for only some selected trees, normally the dominant ones in terms of their DBH. In this study, the two species of trees considered explained thus;

  1. Pinus caribaea:‘Pinus’ is from the Greek word ‘pinos’ (pine tree), possibly from the Celtic term ‘pin’ or ‘pyn’ (mountain or rock), referring to the habitat of the pine.  Pinus caribaea is a fine tree to 20-30m tall, often 35m, with a diameter of 50-80cm and occasionally up to 1m; trunk generally straight and well formed; lower branches large, horizontal and drooping; upper branches often ascending to form an open, rounded to pyramidal crown; young trees with a dense, pyramidal crown. Pinus caribaea is rated as moderately fire resistant. It tolerates salt winds and hence may be planted near the coast.
  2. Bombax costatum:'Bombax' is derived from the Greek 'bombux', meaning silk, alluding to the dense wool-like floss covering the inner walls of the fruits and the seeds. Bombax costatum is a fire resisting tree of the savannas and dry woodlands from Senegal to central Africa, from Guinea across Ghana and Nigeria to southern Chad. Its tuberous roots act as water and/or sugar storage facilities during long drought periods. Usually associated with Pterocarpus erinaceus, Daniellia oliveri, Cordyla pinnata, Parkia biglobosa, Terminalia macroptera and Prosopis africana.

Length-biased and area-biased distribution

When the probability of selecting an individual in a population is proportional to its magnitude, it is called length biased sampling. However, when observations are selected with probability proportional to their length, the resulting distribution is called length-biased. When dealing with the problem of sampling and selection from a length-biased distribution, the possible bias due to the nature of data-collection process can be utilized to connect the population parameters to that of the sampling distribution. That is, biased sampling is not always detrimental to the process of inference on population parameters. Inference based on a biased sample of a certain size may yield more information than that given by an unbiased sample of the same size, provided that the choice mechanism behind the biased sample is known. Statistical analysis based on length-biased samples has been studied in detail since the early 70’s. Size-biased distributions have been found to be useful in probability sampling designs for forestry and other related studies. These designs are classified into length-biased methods where sampling is done with probability proportional to some lineal measure and area-biased methods where units are selected into the sample with probability proportional to some real attributes. Hence, area-biased distribution is the square of the random variable of X or the second order power of size-biased distribution

The concept of length-biased was introduced by Cox in 1962.1 This concept is found in various applications in biomedical area such as family history and disease, survival analysis, intermediate events and latency period of AIDS due to blood transfusion. Many works were done to characterize relationships between original distributions and their length-biased versions. Patill and Rao expressed some basic distributions and their length-biased forms such as log-normal, gamma, pareto, beta distributions. Recently, many researches are applied to length-biased for lifetime distribution, length-biased weighted Weibull distribution, and length-biased weighted generalized Rayleigh distribution, length-biased beta distribution, and Bayes estimation of length- biased Weibull distribution.2

Exponentiated weibull distribution

The Weibull distribution was introduced by Wallodi Weibull, Swedish scientist, in 1951. It is perhaps the most widely used distribution to analyze the lifetime data. Gupta & Kundu3 proposed an Exponentiated Exponential distribution which is a special case of the Exponentiated Weibull family. Flaih et al.,4 extended the Inverted Weibull distribution to the Exponentiated Inverted Weibull (EIW) distribution by adding another shape parameter. This study suggested that the EIW distribution can provide a better fit to the real dataset than the IW distribution. Shittu, O I. and Adepoju, K A.5 the exponentiated Weibull was used as an alternative distribution that adequately describe the wind speed and thereby provide better representation of the potentials of wind energy.

Structural properties of exponentiated weibull distribution: According to Mudhokar, et al.,6 the Exponentiated Weibull density function is defined as;

f EW (x;k,λ,α)= αk x k-1 λ k exp( - (x/λ) k ) ( 1-exp( - (x/λ) k ) ) α-1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaabaGaamyraiaadEfaaeqaaiaacIcacaWG4bGaai4oaiaadUga caGGSaGaeq4UdWMaaiilaiabeg7aHjaacMcacqGH9aqpdaWcaaqaai abeg7aHjaadUgacaWG4bWcdaahaaqcfayabeaajugWaiaadUgacaGG TaGaaGymaaaaaKqbagaacqaH7oaBdaahaaqabeaajugWaiaadUgaaa aaaKqbakGacwgacaGG4bGaaiiCamaabmaabaGaaiylaiaacIcacaWG 4bGaai4laiabeU7aSjaacMcalmaaCaaajuaGbeqaaKqzadGaam4Aaa aaaKqbakaawIcacaGLPaaadaqadaqaaiaaigdacaGGTaGaciyzaiaa cIhacaGGWbWaaeWaaeaacaGGTaGaaiikaiaadIhacaGGVaGaeq4UdW MaaiykaSWaaWbaaKqbagqabaqcLbmacaWGRbaaaaqcfaOaayjkaiaa wMcaaaGaayjkaiaawMcaaSWaaWbaaKqbagqabaqcLbmacqaHXoqyca GGTaGaaGymaaaaaaa@7240@ …… (1)
and the cdf is;
F EW (x;k,λ,α)= ( 1exp( (x/λ) k ) ) α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaabaGaamyraiaadEfaaeqaaiaacIcacaWG4bGaai4oaiaadUga caGGSaGaeq4UdWMaaiilaiabeg7aHjaacMcacqGH9aqpdaqadaqaai aaigdacqGHsislciGGLbGaaiiEaiaacchadaqadaqaaiabgkHiTiaa cIcacaWG4bGaai4laiabeU7aSjaacMcadaahaaqabeaajugWaiaadU gaaaaajuaGcaGLOaGaayzkaaaacaGLOaGaayzkaaWcdaahaaqcfaya beaajugWaiabeg7aHbaaaaa@5669@

α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@  and k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbaaaa@3794@ are shape parameters; λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@  is a scale parameter.
the r th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaeOCam aaCaaabeqaaKqzadGaaeiDaiaabIgaaaaaaa@3AAB@  moment of the exponentiated weibull is given as;
E( x r )=α λ r Γ( r k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) r k +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyrai aacIcacaWG4bWaaWbaaeqabaqcLbmacaWGYbaaaKqbakaacMcacqGH 9aqpcqaHXoqycqaH7oaBdaahaaqabeaajugWaiaadkhaaaqcfaOaeu 4KdC0aaeWaaeaadaWcaaqaaiaadkhaaeaacaWGRbaaaiabgUcaRiaa igdaaiaawIcacaGLPaaadaaeWbqaamaalaaabaGaaiikaiabgkHiTi aaigdacaGGPaWaaWbaaeqabaqcLbmacaWGQbaaaKqbaoaabmaabaWa a0baaeaacaWGQbaabaGaeqySdeMaeyOeI0IaaGymaaaaaiaawIcaca GLPaaaaeaadaqadaqaaiaaigdacqGHRaWkcaWGQbaacaGLOaGaayzk aaWcdaahaaadbeqaamaaCaaabeqaamaalaaabaGaamOCaaqaaiaadU gaaaGaey4kaSIaaGymaaaaaaaaaaqcfayaaKqzadGaamOAaiabg2da 9iaaicdaaKqbagaajugWaiabg6HiLcqcfaOaeyyeIuoaaaa@683F@

Where Γ( r k +1 )= 0 x r k exp(x)x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeu4KdC 0aaeWaaeaadaWcaaqaaiaadkhaaeaacaWGRbaaaiabgUcaRiaaigda aiaawIcacaGLPaaacqGH9aqpdaWdXbqaaiaadIhadaahaaqabeaada WcaaqaaiaadkhaaeaacaWGRbaaaaaaciGGLbGaaiiEaiaacchacaGG OaGaeyOeI0IaamiEaiaacMcacqGHciITcaWG4baabaqcLbmacaaIWa aajuaGbaqcLbmacqGHEisPaKqbakabgUIiYdaaaa@519A@  at r=1, the first moment of EW is

E(x)=αλΓ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaeyrai aabIcacaqG4bGaaiykaiabg2da9iabeg7aHjabeU7aSjabfo5ahnaa bmaabaWaaSaaaeaacaaIXaaabaGaam4AaaaacqGHRaWkcaaIXaaaca GLOaGaayzkaaWaaabCaeaadaWcaaqaaiaacIcacqGHsislcaaIXaGa aiykamaaCaaabeqaaKqzadGaamOAaaaajuaGdaqadaqaamaaDaaaba GaamOAaaqaaiabeg7aHjabgkHiTiaaigdaaaaacaGLOaGaayzkaaaa baWaaeWaaeaacaaIXaGaey4kaSIaamOAaaGaayjkaiaawMcaaSWaaW baaKqbagqabaWcdaWcaaqcfayaaKqzadGaaGymaaqcfayaaKqzadGa am4AaaaacqGHRaWkcaaIXaaaaaaaaKqbagaajugWaiaadQgacqGH9a qpcaaIWaaajuaGbaqcLbmacqGHEisPaKqbakabggHiLdaaaa@65FB@

at r=2 is the second moment and the variance of EW is given thus;

Var(x)=E(x 2 )-[E(x)] 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaeOvai aabggacaqGYbGaaeikaiaabIhacaqGPaGaaeypaiaabweacaqGOaGa aeiEamaaCaaabeqaaKqzadGaaeOmaaaajuaGcaqGPaGaaeylaiaabU facaqGfbGaaeikaiaabIhacaqGPaGaaeyxaSWaaWbaaKqbagqabaqc LbmacaqGYaaaaaaa@4A1B@
Var(x)=α λ 2 Γ( 2 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 2 k +1 α 2 λ 2 Γ 2 ( 1 k +1 ) [ j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 ] 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaeOvai aabggacaqGYbGaaeikaiaabIhacaqGPaGaaeypaiabeg7aHjabeU7a STWaaWbaaKqbagqabaqcLbmacaaIYaaaaKqbakabfo5ahnaabmaaba WaaSaaaeaacaaIYaaabaGaam4AaaaacqGHRaWkcaaIXaaacaGLOaGa ayzkaaWaaabCaeaadaWcaaqaaiaacIcacqGHsislcaaIXaGaaiykam aaCaaabeqaaKqzadGaamOAaaaajuaGdaqadaqaamaaDaaabaGaamOA aaqaaiabeg7aHjabgkHiTiaaigdaaaaacaGLOaGaayzkaaaabaWaae WaaeaacaaIXaGaey4kaSIaamOAaaGaayjkaiaawMcaamaaCaaabeqa aSWaaSaaaKqbagaajugWaiaaikdaaKqbagaajugWaiaadUgaaaGaey 4kaSIaaGymaaaaaaaajuaGbaqcLbmacaWGQbGaeyypa0JaaGimaaqc fayaaKqzadGaeyOhIukajuaGcqGHris5aiabgkHiTiabeg7aHnaaCa aabeqaaKqzadGaaGOmaaaajuaGcqaH7oaBlmaaCaaajuaGbeqaaKqz adGaaGOmaaaajuaGcqqHtoWrdaahaaqabeaajugWaiaaikdaaaqcfa 4aaeWaaeaadaWcaaqaaiaaigdaaeaacaWGRbaaaiabgUcaRiaaigda aiaawIcacaGLPaaadaWadaqaamaaqahabaWaaSaaaeaacaGGOaGaey OeI0IaaGymaiaacMcalmaaCaaajuaGbeqaaKqzadGaamOAaaaajuaG daqadaqaamaaDaaabaGaamOAaaqaaiabeg7aHjabgkHiTiaaigdaaa aacaGLOaGaayzkaaaabaWaaeWaaeaacaaIXaGaey4kaSIaamOAaaGa ayjkaiaawMcaaSWaaWbaaKqbagqabaWcdaWcaaqcfayaaKqzadGaaG ymaaqcfayaaKqzadGaam4AaaaacqGHRaWkcaaIXaaaaaaaaKqbagaa jugWaiaadQgacqGH9aqpcaaIWaaajuaGbaqcLbmacqGHEisPaKqbak abggHiLdaacaGLBbGaayzxaaWcdaahaaqcfayabeaajugWaiaaikda aaaaaa@A414@
The skewness and kurtosis of EW
k 3 = αλΓ( 3 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 3 k +1 [ 3αλΓ( 2 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 2 k +1 ][ αλΓ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 ]+2αλΓ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 [ Var(x) ] 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaae4Aam aaBaaabaqcLbmacaqGZaaajuaGbeaacaqG9aWaaSaaaeaacqaHXoqy cqaH7oaBcqqHtoWrdaqadaqaamaalaaabaGaaG4maaqaaiaadUgaaa Gaey4kaSIaaGymaaGaayjkaiaawMcaamaaqahabaWaaSaaaeaacaGG OaGaeyOeI0IaaGymaiaacMcadaahaaqabKqbGeaacaWGQbaaaKqbao aabmaabaWaa0baaeaacaWGQbaabaGaeqySdeMaeyOeI0IaaGymaaaa aiaawIcacaGLPaaaaeaadaqadaqaaiaaigdacqGHRaWkcaWGQbaaca GLOaGaayzkaaWaaWbaaeqajuaibaqcfa4aaSaaaKqbGeaacaaIZaaa baGaam4AaaaacqGHRaWkcaaIXaaaaaaaaeaacaWGQbGaeyypa0JaaG imaaqcfayaaKqzadGaeyOhIukajuaGcqGHris5aiabgkHiTmaadmaa baGaaG4maiabeg7aHjabeU7aSjabfo5ahnaabmaabaWaaSaaaeaaca aIYaaabaGaam4AaaaacqGHRaWkcaaIXaaacaGLOaGaayzkaaWaaabC aeaadaWcaaqaaiaacIcacqGHsislcaaIXaGaaiykamaaCaaabeqcfa saaiaadQgaaaqcfa4aaeWaaeaadaqhaaqaaiaadQgaaeaacqaHXoqy cqGHsislcaaIXaaaaaGaayjkaiaawMcaaaqaamaabmaabaGaaGymai abgUcaRiaadQgaaiaawIcacaGLPaaadaahaaqabKqbGeaajuaGdaWc aaqcfasaaiaaikdaaeaacaWGRbaaaiabgUcaRiaaigdaaaaaaaqaai aadQgacqGH9aqpcaaIWaaajuaGbaqcLbmacqGHEisPaKqbakabggHi LdaacaGLBbGaayzxaaWaamWaaeaacqaHXoqycqaH7oaBcqqHtoWrda qadaqaamaalaaabaGaaGymaaqaaiaadUgaaaGaey4kaSIaaGymaaGa ayjkaiaawMcaamaaqahabaWaaSaaaeaacaGGOaGaeyOeI0IaaGymai aacMcadaahaaqabKqbGeaacaWGQbaaaKqbaoaabmaabaWaa0baaeaa caWGQbaabaGaeqySdeMaeyOeI0IaaGymaaaaaiaawIcacaGLPaaaae aadaqadaqaaiaaigdacqGHRaWkcaWGQbaacaGLOaGaayzkaaWaaWba aeqajuaibaqcfa4aaSaaaKqbGeaacaaIXaaabaGaam4AaaaacqGHRa WkcaaIXaaaaaaaaeaacaWGQbGaeyypa0JaaGimaaqcfayaaKqzadGa eyOhIukajuaGcqGHris5aaGaay5waiaaw2faaiabgUcaRiaaikdacq aHXoqycqaH7oaBcqqHtoWrdaqadaqaamaalaaabaGaaGymaaqaaiaa dUgaaaGaey4kaSIaaGymaaGaayjkaiaawMcaamaaqahabaWaaSaaae aacaGGOaGaeyOeI0IaaGymaiaacMcadaahaaqabKqbGeaacaWGQbaa aKqbaoaabmaabaWaa0baaeaacaWGQbaabaGaeqySdeMaeyOeI0IaaG ymaaaaaiaawIcacaGLPaaaaeaadaqadaqaaiaaigdacqGHRaWkcaWG QbaacaGLOaGaayzkaaWaaWbaaeqajuaibaqcfa4aaSaaaKqbGeaaca aIXaaabaGaam4AaaaacqGHRaWkcaaIXaaaaaaaaeaacaWGQbGaeyyp a0JaaGimaaqcfayaaKqzadGaeyOhIukajuaGcqGHris5aaqaamaadm aabaGaaeOvaiaabggacaqGYbGaaeikaiaabIhacaqGPaaacaGLBbGa ayzxaaWcdaahaaqcfayabeaajugWaiaaiodacaGGVaGaaGOmaaaaaa aaaa@E2B2@

Materials and methods

In this study, we propose two new distributions which are LBEW and ABEW distributions. We first provide a general definition of the Length-biased and Area-biased distributions which we subsequently reveal their pdfs.
Let f(x;θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzai aacIcacaWG4bGaai4oaiabeI7aXjaacMcaaaa@3C3A@ be the pdf of the random variable X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbIt LDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEe eu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=J b9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabau aaaOqaaKqbacbaaaaaaaaapeGaamiwaaaa@3B89@  and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  be the unknown parameter. The weighted distribution is defined as;

k 4 = αλΓ( 4 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 4 k +1 [ 4αλΓ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 ][ αλΓ( 3 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 3 k +1 ]+[ 6 α 2 λΓ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 2 k +1 ][ αλΓ( 2 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 2 k +1 ]3 α 4 λΓ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 [ Var(x) ] 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaae4AaS WaaSbaaKqbagaajugWaiaabsdaaKqbagqaaiaab2dadaWcaaqaaiab eg7aHjabeU7aSjabfo5ahnaabmaabaWaaSaaaeaacaaI0aaabaGaam 4AaaaacqGHRaWkcaaIXaaacaGLOaGaayzkaaWaaabCaeaadaWcaaqa aiaacIcacqGHsislcaaIXaGaaiykamaaCaaabeqaaKqzadGaamOAaa aajuaGdaqadaqaamaaDaaabaGaamOAaaqaaiabeg7aHjabgkHiTiaa igdaaaaacaGLOaGaayzkaaaabaWaaeWaaeaacaaIXaGaey4kaSIaam OAaaGaayjkaiaawMcaamaaCaaabeqaaSWaaSaaaKqbagaajugWaiaa isdaaKqbagaajugWaiaadUgaaaGaey4kaSIaaGymaaaaaaaajuaGba qcLbmacaWGQbGaeyypa0JaaGimaaqcfayaaKqzadGaeyOhIukajuaG cqGHris5aiabgkHiTmaadmaabaGaaGinaiabeg7aHjabeU7aSjabfo 5ahnaabmaabaWaaSaaaeaacaaIXaaabaGaam4AaaaacqGHRaWkcaaI XaaacaGLOaGaayzkaaWaaabCaeaadaWcaaqaaiaacIcacqGHsislca aIXaGaaiykamaaCaaabeqaaKqzadGaamOAaaaajuaGdaqadaqaamaa DaaabaGaamOAaaqaaiabeg7aHjabgkHiTiaaigdaaaaacaGLOaGaay zkaaaabaWaaeWaaeaacaaIXaGaey4kaSIaamOAaaGaayjkaiaawMca amaaCaaabeqaaSWaaSaaaKqbagaajugWaiaaigdaaKqbagaajugWai aadUgaaaGaey4kaSIaaGymaaaaaaaajuaGbaqcLbmacaWGQbGaeyyp a0JaaGimaaqcfayaaKqzadGaeyOhIukajuaGcqGHris5aaGaay5wai aaw2faamaadmaabaGaeqySdeMaeq4UdWMaeu4KdC0aaeWaaeaadaWc aaqaaiaaiodaaeaacaWGRbaaaiabgUcaRiaaigdaaiaawIcacaGLPa aadaaeWbqaamaalaaabaGaaiikaiabgkHiTiaaigdacaGGPaWaaWba aeqabaqcLbmacaWGQbaaaKqbaoaabmaabaWaa0baaeaacaWGQbaaba GaeqySdeMaeyOeI0IaaGymaaaaaiaawIcacaGLPaaaaeaadaqadaqa aiaaigdacqGHRaWkcaWGQbaacaGLOaGaayzkaaWaaWbaaeqabaWcda WcaaqcfayaaKqzadGaaG4maaqcfayaaKqzadGaam4AaaaacqGHRaWk caaIXaaaaaaaaKqbagaajugWaiaadQgacqGH9aqpcaaIWaaajuaGba qcLbmacqGHEisPaKqbakabggHiLdaacaGLBbGaayzxaaGaey4kaSYa amWaaeaacaaI2aGaeqySde2aaWbaaeqabaGaaGOmaaaacqaH7oaBcq qHtoWrdaqadaqaamaalaaabaGaaGymaaqaaiaadUgaaaGaey4kaSIa aGymaaGaayjkaiaawMcaamaaqahabaWaaSaaaeaacaGGOaGaeyOeI0 IaaGymaiaacMcadaahaaqabeaajugWaiaadQgaaaqcfa4aaeWaaeaa daqhaaqaaiaadQgaaeaacqaHXoqycqGHsislcaaIXaaaaaGaayjkai aawMcaaaqaamaabmaabaGaaGymaiabgUcaRiaadQgaaiaawIcacaGL PaaadaahaaqabeaalmaalaaajuaGbaqcLbmacaaIYaaajuaGbaqcLb macaWGRbaaaiabgUcaRiaaigdaaaaaaaqcfayaaKqzadGaamOAaiab g2da9iaaicdaaKqbagaajugWaiabg6HiLcqcfaOaeyyeIuoaaiaawU facaGLDbaadaWadaqaaiabeg7aHjabeU7aSjabfo5ahnaabmaabaWa aSaaaeaacaaIYaaabaGaam4AaaaacqGHRaWkcaaIXaaacaGLOaGaay zkaaWaaabCaeaadaWcaaqaaiaacIcacqGHsislcaaIXaGaaiykamaa CaaabeqaaKqzadGaamOAaaaajuaGdaqadaqaamaaDaaabaGaamOAaa qaaiabeg7aHjabgkHiTiaaigdaaaaacaGLOaGaayzkaaaabaWaaeWa aeaacaaIXaGaey4kaSIaamOAaaGaayjkaiaawMcaamaaCaaabeqaaS WaaSaaaKqbagaajugWaiaaikdaaKqbagaajugWaiaadUgaaaGaey4k aSIaaGymaaaaaaaajuaGbaqcLbmacaWGQbGaeyypa0JaaGimaaqcfa yaaKqzadGaeyOhIukajuaGcqGHris5aaGaay5waiaaw2faaiabgkHi TiaaiodacqaHXoqydaahaaqabeaacaaI0aaaaiabeU7aSjabfo5ahn aabmaabaWaaSaaaeaacaaIXaaabaGaam4AaaaacqGHRaWkcaaIXaaa caGLOaGaayzkaaWaaabCaeaadaWcaaqaaiaacIcacqGHsislcaaIXa GaaiykamaaCaaabeqaaKqzadGaamOAaaaajuaGdaqadaqaamaaDaaa baGaamOAaaqaaiabeg7aHjabgkHiTiaaigdaaaaacaGLOaGaayzkaa aabaWaaeWaaeaacaaIXaGaey4kaSIaamOAaaGaayjkaiaawMcaamaa CaaabeqaaSWaaSaaaKqbagaajugWaiaaigdaaKqbagaajugWaiaadU gaaaGaey4kaSIaaGymaaaaaaaajuaGbaqcLbmacaWGQbGaeyypa0Ja aGimaaqcfayaaKqzadGaeyOhIukajuaGcqGHris5aaqaamaadmaaba GaaeOvaiaabggacaqGYbGaaeikaiaabIhacaqGPaaacaGLBbGaayzx aaWcdaahaaqcfayabeaajugWaiaaikdaaaaaaaaa@5402@

g(x;θ)= x m f(x;θ) E[f(x)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbIt LDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEe eu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=J b9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabau aaaOqaaKqbakaadEgacaGGOaGaamiEaiaacUdacqaH4oqCcaGGPaGa eyypa0ZaaSaaaeaacaWG4bWaaWbaaeqabaqcLbmacaWGTbaaaKqbak aadAgacaGGOaGaamiEaiaacUdacqaH4oqCcaGGPaaabaGaamyraiaa cUfacaWGMbGaaiikaiaadIhacaGGPaGaaiyxaaaaaaa@50A7@ XR, θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbIt LDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEe eu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=J b9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabau aaaOabaeqabaqcfaOaamiwaiabgIGiolaadkfacaGGSaaabaGaeqiU deNaeyOpa4JaaGimaaaaaa@41F3@ ………….. (2)
The distributions in equation (2) are termed as size-biased distribution of order m. When m=1, it is called size-biased of order 1 or say length biased distribution, whereas for m=2, it is called the area- biased distribution.

Length-biased EW distribution (LBEW)

If X has a lifetime distribution with pdf f( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbIt LDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEe eu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=J b9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabau aaaOqaaKqbacbaaaaaaaaapeGaamOza8aadaqadaqaa8qacaWG4baa paGaayjkaiaawMcaaaaa@3E4B@ and expected value, E[ f( x ) ]< MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbIt LDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEe eu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=J b9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabau aaaOqaaKqbacbaaaaaaaaapeGaamyramaadmaabaGaamOza8aadaqa daqaa8qacaWG4baapaGaayjkaiaawMcaaaWdbiaawUfacaGLDbaacq GH8aapcqGHEisPaaa@438C@ , the pdf of length-biased distribution of X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGybaaaa@3781@  can be defined as:
g LB (x;θ)= xf(x;θ) E[f(x)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4zam aaBaaabaGaamitaiaadkeaaeqaaiaacIcacaWG4bGaai4oaiabeI7a XjaacMcacqGH9aqpdaWcaaqaaiaadIhacaWGMbGaaiikaiaadIhaca GG7aGaeqiUdeNaaiykaaqaaiaadweacaGGBbGaamOzaiaacIcacaWG 4bGaaiykaiaac2faaaaaaa@4B88@ ………….. (3)

Let X be a random variable of an EW distribution with pdf f( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbIt LDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEe eu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=J b9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabau aaaOqaaKqbacbaaaaaaaaapeGaamOza8aadaqadaqaa8qacaWG4baa paGaayjkaiaawMcaaaaa@3E4B@ .

Then g LB (x;θ)= xf(x;θ) E[f(x)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4zam aaBaaabaGaamitaiaadkeaaeqaaiaacIcacaWG4bGaai4oaiabeI7a XjaacMcacqGH9aqpdaWcaaqaaiaadIhacaWGMbGaaiikaiaadIhaca GG7aGaeqiUdeNaaiykaaqaaiaadweacaGGBbGaamOzaiaacIcacaWG 4bGaaiykaiaac2faaaaaaa@4B88@ is a pdf of the LBEW distribution with two shape parameters α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3795@ and k and a scale parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWgaaa@37AA@ . The notation for X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGybaaaa@3781@ with the LBEW distribution is denoted as X ~LBEW ( α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ , k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbaaaa@3794@ , λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ ). The pdf of X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGybaaaa@3781@ is given by:

g LBEW (x;k,λ,α)= k x k λ k+1 exp( (x/λ) k ) ( 1exp( (x/λ) k ) ) α1 Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4zam aaBaaabaqcLbmacaWGmbGaamOqaiaadweacaWGxbaajuaGbeaacaGG OaGaamiEaiaacUdacaWGRbGaaiilaiabeU7aSjaacYcacqaHXoqyca GGPaGaeyypa0ZaaSaaaeaadaWcaaqaaiaadUgacaWG4bWaaWbaaeqa baqcLbmacaWGRbaaaaqcfayaaiabeU7aSnaaCaaabeqaaKqzadGaam 4AaiabgUcaRiaaigdaaaaaaKqbakGacwgacaGG4bGaaiiCamaabmaa baGaeyOeI0IaaiikaiaadIhacaGGVaGaeq4UdWMaaiykamaaCaaabe qaaKqzadGaam4AaaaaaKqbakaawIcacaGLPaaadaqadaqaaiaaigda cqGHsislciGGLbGaaiiEaiaacchadaqadaqaaiabgkHiTiaacIcaca WG4bGaai4laiabeU7aSjaacMcadaahaaqabeaajugWaiaadUgaaaaa juaGcaGLOaGaayzkaaaacaGLOaGaayzkaaWaaWbaaeqabaqcLbmacq aHXoqycqGHsislcaaIXaaaaaqcfayaaiabfo5ahnaabmaabaWaaSaa aeaacaaIXaaabaGaam4AaaaacqGHRaWkcaaIXaaacaGLOaGaayzkaa WaaabCaeaadaWcaaqaaiaacIcacqGHsislcaaIXaGaaiykamaaCaaa beqaaKqzadGaamOAaaaajuaGdaqadaqaaSWaa0baaKqbagaajugWai aadQgaaKqbagaajugWaiabeg7aHjabgkHiTiaaigdaaaaajuaGcaGL OaGaayzkaaaabaWaaeWaaeaacaaIXaGaey4kaSIaamOAaaGaayjkai aawMcaamaaCaaabeqaaSWaaSaaaKqbagaajugWaiaaigdaaKqbagaa jugWaiaadUgaaaGaey4kaSIaaGymaaaaaaaajuaGbaqcLbmacaWGQb Gaeyypa0JaaGimaaGcbaqcLbmacqGHEisPaKqbakabggHiLdaaaaaa @9E48@ ………….. (4)

Area-biased EW distribution (ABEW)

If X has a lifetime distribution with pdf f( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGMbWdamaabmaabaWdbiaadIhaa8aacaGLOaGaayzkaaaa aa@3A43@ and expected value, E[ f( X ) ] < MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGfbWdamaadmaabaWdbiaadAgapaWaaeWaaeaapeGaamiw aaWdaiaawIcacaGLPaaaaiaawUfacaGLDbaapeGaaeiiaiabgYda8i abg6HiLcaa@4026@ , the pdf of length-biased distribution of X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGybaaaa@3781@  can be defined as:

g AB (x;θ)= x 2 f(x;θ) E[f(x)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4zam aaBaaabaqcLbmacaWGbbGaamOqaaqcfayabaGaaiikaiaadIhacaGG 7aGaeqiUdeNaaiykaiabg2da9maalaaabaGaamiEamaaCaaabeqaaK qzadGaaGOmaaaajuaGcaWGMbGaaiikaiaadIhacaGG7aGaeqiUdeNa aiykaaqaaiaadweacaGGBbGaamOzaiaacIcacaWG4bGaaiykaiaac2 faaaaaaa@4FD3@ ………….. (5)

Let X be a random variable of an EW distribution with pdf f( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGMbWdamaabmaabaWdbiaadIhaa8aacaGLOaGaayzkaaaa aa@3A43@ . Then g AB (x;θ)= x 2 f(x;θ) E[f(x)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4zam aaBaaabaqcLbmacaWGbbGaamOqaaqcfayabaGaaiikaiaadIhacaGG 7aGaeqiUdeNaaiykaiabg2da9maalaaabaGaamiEamaaCaaabeqaaK qzadGaaGOmaaaajuaGcaWGMbGaaiikaiaadIhacaGG7aGaeqiUdeNa aiykaaqaaiaadweacaGGBbGaamOzaiaacIcacaWG4bGaaiykaiaac2 faaaaaaa@4FD3@ is a pdf of the ABEW distribution with two shape parameters α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ and k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbaaaa@3794@ and a scale parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ . The notation for X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGybaaaa@3781@ with the ABEW distribution is denoted as X~ABEW MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGybGaaiOFaiaadgeacaWGcbGaamyraiaadEfaaaa@3BB6@  ( α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ , k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbaaaa@3794@ , λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ ). The pdf of X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGybaaaa@3781@ is given by:

g ABEW (x;k,λ,α)= k x k+1 λ k+1 exp( (x/λ) k ) ( 1exp( (x/λ) k ) ) α1 Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4zam aaBaaabaqcLbmacaWGbbGaamOqaiaadweacaWGxbaajuaGbeaacaGG OaGaamiEaiaacUdacaWGRbGaaiilaiabeU7aSjaacYcacqaHXoqyca GGPaGaeyypa0ZaaSaaaeaadaWcaaqaaiaadUgacaWG4bWaaWbaaeqa baqcLbmacaWGRbGaey4kaSIaaGymaaaaaKqbagaacqaH7oaBdaahaa qabeaajugWaiaadUgacqGHRaWkcaaIXaaaaaaajuaGciGGLbGaaiiE aiaacchadaqadaqaaiabgkHiTiaacIcacaWG4bGaai4laiabeU7aSj aacMcalmaaCaaajuaGbeqaaKqzadGaam4AaaaaaKqbakaawIcacaGL PaaadaqadaqaaiaaigdacqGHsislciGGLbGaaiiEaiaacchadaqada qaaiabgkHiTiaacIcacaWG4bGaai4laiabeU7aSjaacMcadaahaaqa beaacaWGRbaaaaGaayjkaiaawMcaaaGaayjkaiaawMcaaSWaaWbaaK qbagqabaqcLbmacqaHXoqycqGHsislcaaIXaaaaaqcfayaaiabfo5a hnaabmaabaWaaSaaaeaacaaIXaaabaGaam4AaaaacqGHRaWkcaaIXa aacaGLOaGaayzkaaWaaabCaeaadaWcaaqaaiaacIcacqGHsislcaaI XaGaaiykamaaCaaabeqaaKqzadGaamOAaaaajuaGdaqadaqaamaaDa aabaGaamOAaaqaaiabeg7aHjabgkHiTiaaigdaaaaacaGLOaGaayzk aaaabaWaaeWaaeaacaaIXaGaey4kaSIaamOAaaGaayjkaiaawMcaaS WaaWbaaKqbagqabaWcdaWcaaqcfayaaKqzadGaaGymaaqcfayaaKqz adGaam4AaaaacqGHRaWkcaaIXaaaaaaaaKqbagaajugWaiaadQgacq GH9aqpcaaIWaaajuaGbaqcLbmacqGHEisPaKqbakabggHiLdaaaaaa @9C5C@ ………….. (6)

The properties

The LBEW distribution properties are as follows;

0 g LBEW (x) x= 0 k x k λ k+1 exp( (x/λ) k ) ( 1exp( (x/λ) k ) ) α1 Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCae aajugWaiaadEgalmaaBaaajuaGbaqcLbmacaWGmbGaamOqaiaadwea caWGxbaajuaGbeaacaGGOaGaamiEaiaacMcaaeaajugWaiaaicdaaK qbagaajugWaiabg6HiLcqcfaOaey4kIipacqGHciITcaWG4bGaeyyp a0Zaa8qCaeaadaWcaaqaamaalaaabaGaam4AaiaadIhadaahaaqabe aajugWaiaadUgaaaaajuaGbaGaeq4UdW2cdaahaaqcfayabeaajugW aiaadUgacqGHRaWkcaaIXaaaaaaajuaGciGGLbGaaiiEaiaacchada qadaqaaiabgkHiTiaacIcacaWG4bGaai4laiabeU7aSjaacMcalmaa CaaajuaGbeqaaKqzadGaam4AaaaaaKqbakaawIcacaGLPaaadaqada qaaiaaigdacqGHsislciGGLbGaaiiEaiaacchadaqadaqaaiabgkHi TiaacIcacaWG4bGaai4laiabeU7aSjaacMcadaahaaqabeaajugWai aadUgaaaaajuaGcaGLOaGaayzkaaaacaGLOaGaayzkaaWcdaahaaqc fayabeaajugWaiabeg7aHjabgkHiTiaaigdaaaaajuaGbaGaeu4KdC 0aaeWaaeaadaWcaaqaaiaaigdaaeaacaWGRbaaaiabgUcaRiaaigda aiaawIcacaGLPaaadaaeWbqaamaalaaabaGaaiikaiabgkHiTiaaig dacaGGPaWcdaahaaqcfayabeaajugWaiaadQgaaaqcfa4aaeWaaeaa daqhaaqaaiaadQgaaeaacqaHXoqycqGHsislcaaIXaaaaaGaayjkai aawMcaaaqaamaabmaabaGaaGymaiabgUcaRiaadQgaaiaawIcacaGL PaaadaahaaqabeaalmaalaaajuaGbaqcLbmacaaIXaaajuaGbaqcLb macaWGRbaaaiabgUcaRiaaigdaaaaaaaqcfayaaKqzadGaamOAaiab g2da9iaaicdaaKqbagaajugWaiabg6HiLcqcfaOaeyyeIuoaaaGaey OaIyRaamiEaaqaaKqzadGaaGimaaqcfayaaKqzadGaeyOhIukajuaG cqGHRiI8aaaa@AD32@
= k λ k+1 Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 0 x k exp( (x/λ) k ) ( 1exp( (x/λ) k ) ) α1 x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaadaWcaaqaaiaadUgaaeaacqaH7oaBdaahaaqabeaajugW aiaadUgacqGHRaWkcaaIXaaaaaaaaKqbagaacqqHtoWrdaqadaqaam aalaaabaGaaGymaaqaaiaadUgaaaGaey4kaSIaaGymaaGaayjkaiaa wMcaamaaqahabaWaaSaaaeaacaGGOaGaeyOeI0IaaGymaiaacMcada ahaaqabeaacaWGQbaaamaabmaabaWaa0baaeaacaWGQbaabaGaeqyS deMaeyOeI0IaaGymaaaaaiaawIcacaGLPaaaaeaadaqadaqaaiaaig dacqGHRaWkcaWGQbaacaGLOaGaayzkaaWcdaahaaqcfayabeaalmaa laaajuaGbaqcLbmacaaIXaaajuaGbaqcLbmacaWGRbaaaiabgUcaRi aaigdaaaaaaaqcfayaaKqzadGaamOAaiabg2da9iaaicdaaKqbagaa jugWaiabg6HiLcqcfaOaeyyeIuoaaaWaa8qCaeaacaWG4bWaaWbaae qabaqcLbmacaWGRbaaaKqbakGacwgacaGG4bGaaiiCamaabmaabaGa eyOeI0IaaiikaiaadIhacaGGVaGaeq4UdWMaaiykamaaCaaabeqaaK qzadGaam4AaaaaaKqbakaawIcacaGLPaaadaqadaqaaiaaigdacqGH sislciGGLbGaaiiEaiaacchadaqadaqaaiabgkHiTiaacIcacaWG4b Gaai4laiabeU7aSjaacMcalmaaCaaajuaGbeqaaKqzadGaam4Aaaaa aKqbakaawIcacaGLPaaaaiaawIcacaGLPaaalmaaCaaajuaGbeqaaK qzadGaeqySdeMaeyOeI0IaaGymaaaajuaGcqGHciITcaWG4baabaqc LbmacaaIWaaajuaGbaqcLbmacqGHEisPaKqbakabgUIiYdaaaa@9662@
If y= (x/λ) k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEai abg2da9iaacIcacaWG4bGaai4laiabeU7aSjaacMcalmaaCaaajuaG beqaaKqzadGaam4Aaaaaaaa@401E@ , then we have that;
= 1 Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 0 y 1 k +2 exp( y ) ( 1exp( y ) ) α1 y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaaIXaaabaGaeu4KdC0aaeWaaeaadaWcaaqaaiaaigda aeaacaWGRbaaaiabgUcaRiaaigdaaiaawIcacaGLPaaadaaeWbqaam aalaaabaGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaeqabaqcLbma caWGQbaaaKqbaoaabmaabaWaa0baaeaacaWGQbaabaGaeqySdeMaey OeI0IaaGymaaaaaiaawIcacaGLPaaaaeaadaqadaqaaiaaigdacqGH RaWkcaWGQbaacaGLOaGaayzkaaWcdaahaaqcfayabeaalmaalaaaju aGbaqcLbmacaaIXaaajuaGbaqcLbmacaWGRbaaaiabgUcaRiaaigda aaaaaaqcfayaaKqzadGaamOAaiabg2da9iaaicdaaKqbagaajugWai abg6HiLcqcfaOaeyyeIuoaaaWaa8qCaeaacaWG5bWcdaahaaqcfaya beaalmaalaaajuaGbaqcLbmacaaIXaaajuaGbaqcLbmacaWGRbaaai abgUcaRiaaikdaaaqcfaOaciyzaiaacIhacaGGWbWaaeWaaeaacqGH sislcaWG5baacaGLOaGaayzkaaWaaeWaaeaacaaIXaGaeyOeI0Iaci yzaiaacIhacaGGWbWaaeWaaeaacqGHsislcaWG5baacaGLOaGaayzk aaaacaGLOaGaayzkaaWcdaahaaqcfayabeaajugWaiabeg7aHjabgk HiTiaaigdaaaqcfaOaeyOaIyRaamyEaaqaaKqzadGaaGimaaqcfaya aKqzadGaeyOhIukajuaGcqGHRiI8aaaa@8960@
Recall that,
( 1exp( y ) ) α1 = j=0 (1) j ( j α1 ) exp( yj ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIXaGaeyOeI0IaciyzaiaacIhacaGGWbWaaeWaaeaacqGHsisl caWG5baacaGLOaGaayzkaaaacaGLOaGaayzkaaWaaWbaaeqabaqcLb macqaHXoqycqGHsislcaaIXaaaaKqbakabg2da9maaqahabaGaaiik aiabgkHiTiaaigdacaGGPaWcdaahaaqcfayabeaajugWaiaadQgaaa qcfa4aaeWaaeaadaqhaaqaaiaadQgaaeaacqaHXoqycqGHsislcaaI XaaaaaGaayjkaiaawMcaaaqaaKqzadGaamOAaiabg2da9iaaicdaaK qbagaajugWaiabg6HiLcqcfaOaeyyeIuoaciGGLbGaaiiEaiaaccha daqadaqaaiabgkHiTiaadMhacaWGQbaacaGLOaGaayzkaaaaaa@6377@
Therefore,
0 g LBEW (x) x= 1 Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 0 y 1 k +2 j=0 (1) j ( j α1 ) exp( y[ 1+j ] )y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCae aacaWGNbWcdaWgaaqcfayaaKqzadGaamitaiaadkeacaWGfbGaam4v aaqcfayabaGaaiikaiaadIhacaGGPaaabaqcLbmacaaIWaaajuaGba qcLbmacqGHEisPaKqbakabgUIiYdGaeyOaIyRaamiEaiabg2da9maa laaabaGaaGymaaqaaiabfo5ahnaabmaabaWaaSaaaeaacaaIXaaaba Gaam4AaaaacqGHRaWkcaaIXaaacaGLOaGaayzkaaWaaabCaeaadaWc aaqaaiaacIcacqGHsislcaaIXaGaaiykamaaCaaabeqaaKqzadGaam OAaaaajuaGdaqadaqaamaaDaaabaGaamOAaaqaaiabeg7aHjabgkHi TiaaigdaaaaacaGLOaGaayzkaaaabaWaaeWaaeaacaaIXaGaey4kaS IaamOAaaGaayjkaiaawMcaamaaCaaabeqaaSWaaSaaaKqbagaajugW aiaaigdaaKqbagaajugWaiaadUgaaaGaey4kaSIaaGymaaaaaaaaju aGbaqcLbmacaWGQbGaeyypa0JaaGimaaqcfayaaKqzadGaeyOhIuka juaGcqGHris5aaaadaWdXbqaaiaadMhalmaaCaaajuaGbeqaaSWaaS aaaKqbagaajugWaiaaigdaaKqbagaajugWaiaadUgaaaGaey4kaSIa aGOmaaaajuaGdaaeWbqaaiaacIcacqGHsislcaaIXaGaaiykaSWaaW baaKqbagqabaqcLbmacaWGQbaaaKqbaoaabmaabaWaa0baaeaacaWG QbaabaGaeqySdeMaeyOeI0IaaGymaaaaaiaawIcacaGLPaaaaeaaju gWaiaadQgacqGH9aqpcaaIWaaajuaGbaqcLbmacqGHEisPaKqbakab ggHiLdGaciyzaiaacIhacaGGWbWaaeWaaeaacqGHsislcaWG5bWaam WaaeaacaaIXaGaey4kaSIaamOAaaGaay5waiaaw2faaaGaayjkaiaa wMcaaiabgkGi2kaadMhaaeaajugWaiaaicdaaKqbagaajugWaiabg6 HiLcqcfaOaey4kIipaaaa@A75E@
; where m =y[ 1+j ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGTbGaaeiiaiabg2da98aacaWG5bWaamWaaeaacaaIXaGa ey4kaSIaamOAaaGaay5waiaaw2faaaaa@3ECA@
= 1 Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 [ j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 ] 0 m 1 k +2 exp( m )m 0 g LBEW (x) x= 1 Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 [ j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 ]Γ( 1 k +1 ) 0 g LBEW (x) x=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGcq GH9aqpdaWcaaqaaiaaigdaaeaacqqHtoWrdaqadaqaamaalaaabaGa aGymaaqaaiaadUgaaaGaey4kaSIaaGymaaGaayjkaiaawMcaamaaqa habaWaaSaaaeaacaGGOaGaeyOeI0IaaGymaiaacMcadaahaaqabeaa jugWaiaadQgaaaqcfa4aaeWaaeaadaqhaaqaaiaadQgaaeaacqaHXo qycqGHsislcaaIXaaaaaGaayjkaiaawMcaaaqaamaabmaabaGaaGym aiabgUcaRiaadQgaaiaawIcacaGLPaaadaahaaqabeaalmaalaaaju aGbaqcLbmacaaIXaaajuaGbaqcLbmacaWGRbaaaiabgUcaRiaaigda aaaaaaqcfayaaKqzadGaamOAaiabg2da9iaaicdaaKqbagaajugWai abg6HiLcqcfaOaeyyeIuoaaaWaamWaaeaadaaeWbqaamaalaaabaGa aiikaiabgkHiTiaaigdacaGGPaWaaWbaaeqabaqcLbmacaWGQbaaaK qbaoaabmaabaWaa0baaeaacaWGQbaabaGaeqySdeMaeyOeI0IaaGym aaaaaiaawIcacaGLPaaaaeaadaqadaqaaiaaigdacqGHRaWkcaWGQb aacaGLOaGaayzkaaWcdaahaaqcfayabeaalmaalaaajuaGbaqcLbma caaIXaaajuaGbaqcLbmacaWGRbaaaiabgUcaRiaaigdaaaaaaaqcfa yaaKqzadGaamOAaiabg2da9iaaicdaaOqaaKqzadGaeyOhIukajuaG cqGHris5aaGaay5waiaaw2faamaapehabaGaamyBaSWaaWbaaKqbag qabaWcdaWcaaqcfayaaKqzadGaaGymaaqcfayaaKqzadGaam4Aaaaa cqGHRaWkcaaIYaaaaKqbakGacwgacaGG4bGaaiiCamaabmaabaGaey OeI0IaamyBaaGaayjkaiaawMcaaiabgkGi2kaad2gaaeaajugWaiaa icdaaKqbagaajugWaiabg6HiLcqcfaOaey4kIipaaeaadaWdXbqaai aadEgalmaaBaaajuaGbaqcLbmacaWGmbGaamOqaiaadweacaWGxbaa juaGbeaacaGGOaGaamiEaiaacMcaaeaajugWaiaaicdaaKqbagaaju gWaiabg6HiLcqcfaOaey4kIipacqGHciITcaWG4bGaeyypa0ZaaSaa aeaacaaIXaaabaGaeu4KdC0aaeWaaeaadaWcaaqaaiaaigdaaeaaca WGRbaaaiabgUcaRiaaigdaaiaawIcacaGLPaaadaaeWbqaamaalaaa baGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaeqabaqcLbmacaWGQb aaaKqbaoaabmaabaWaa0baaeaacaWGQbaabaGaeqySdeMaeyOeI0Ia aGymaaaaaiaawIcacaGLPaaaaeaadaqadaqaaiaaigdacqGHRaWkca WGQbaacaGLOaGaayzkaaWcdaahaaqcfayabeaalmaalaaajuaGbaqc LbmacaaIXaaajuaGbaqcLbmacaWGRbaaaiabgUcaRiaaigdaaaaaaa qcfayaaKqzadGaamOAaiabg2da9iaaicdaaKqbagaajugWaiabg6Hi LcqcfaOaeyyeIuoaaaWaamWaaeaadaaeWbqaamaalaaabaGaaiikai abgkHiTiaaigdacaGGPaWcdaahaaqcfayabeaajugWaiaadQgaaaqc fa4aaeWaaeaadaqhaaqaaiaadQgaaeaacqaHXoqycqGHsislcaaIXa aaaaGaayjkaiaawMcaaaqaamaabmaabaGaaGymaiabgUcaRiaadQga aiaawIcacaGLPaaalmaaCaaajuaGbeqaaSWaaSaaaKqbagaajugWai aaigdaaKqbagaajugWaiaadUgaaaGaey4kaSIaaGymaaaaaaaajuaG baqcLbmacaWGQbGaeyypa0JaaGimaaqcfayaaKqzadGaeyOhIukaju aGcqGHris5aaGaay5waiaaw2faaiabfo5ahnaabmaabaWaaSaaaeaa caaIXaaabaGaam4AaaaacqGHRaWkcaaIXaaacaGLOaGaayzkaaaaba Waa8qCaeaacaWGNbWaaSbaaeaajugWaiaadYeacaWGcbGaamyraiaa dEfaaKqbagqaaiaacIcacaWG4bGaaiykaaqaaKqzadGaaGimaaqcfa yaaKqzadGaeyOhIukajuaGcqGHRiI8aiabgkGi2kaadIhacqGH9aqp caaIXaaaaaa@18B9@

Therefore, the pdf of LBEW distribution sum to 1. NB: It was also obtainable for the ABEW distribution.
The cdf of LBEW, corresponding to (4) is obtained by

F LBEW (X)= 0 x k x k λ k+1 exp( (x/λ) k ) ( 1exp( (x/λ) k ) ) α1 Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaabaqcLbmacaWGmbGaamOqaiaadweacaWGxbaajuaGbeaacaGG OaGaamiwaiaacMcacqGH9aqpdaWdXbqaamaalaaabaWaaSaaaeaaca WGRbGaamiEaSWaaWbaaKqbagqabaqcLbmacaWGRbaaaaqcfayaaiab eU7aSnaaCaaabeqaaKqzadGaam4AaiabgUcaRiaaigdaaaaaaKqbak GacwgacaGG4bGaaiiCamaabmaabaGaeyOeI0IaaiikaiaadIhacaGG VaGaeq4UdWMaaiykamaaCaaabeqaaKqzadGaam4AaaaaaKqbakaawI cacaGLPaaadaqadaqaaiaaigdacqGHsislciGGLbGaaiiEaiaaccha daqadaqaaiabgkHiTiaacIcacaWG4bGaai4laiabeU7aSjaacMcada ahaaqabeaajugWaiaadUgaaaaajuaGcaGLOaGaayzkaaaacaGLOaGa ayzkaaWcdaahaaqcfayabeaajugWaiabeg7aHjabgkHiTiaaigdaaa aajuaGbaGaeu4KdC0aaeWaaeaadaWcaaqaaiaaigdaaeaacaWGRbaa aiabgUcaRiaaigdaaiaawIcacaGLPaaadaaeWbqaamaalaaabaGaai ikaiabgkHiTiaaigdacaGGPaWaaWbaaeqabaGaamOAaaaadaqadaqa amaaDaaabaGaamOAaaqaaiabeg7aHjabgkHiTiaaigdaaaaacaGLOa GaayzkaaaabaWaaeWaaeaacaaIXaGaey4kaSIaamOAaaGaayjkaiaa wMcaamaaCaaabeqaaSWaaSaaaKqbagaajugWaiaaigdaaKqbagaaju gWaiaadUgaaaGaey4kaSIaaGymaaaaaaaajuaGbaqcLbmacaWGQbGa eyypa0JaaGimaaqcfayaaKqzadGaeyOhIukajuaGcqGHris5aaaacq GHciITcaWG4baabaqcLbmacaaIWaaajuaGbaqcLbmacaWG4baajuaG cqGHRiI8aaaa@9D71@
F LBEW (X)= k λ k+1 Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 0 x x k exp( (x/λ) k ) ( 1exp( (x/λ) k ) ) α1 x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaabaqcLbmacaWGmbGaamOqaiaadweacaWGxbaajuaGbeaacaGG OaGaamiwaiaacMcacqGH9aqpdaWcaaqaamaalaaabaGaam4Aaaqaai abeU7aSTWaaWbaaKqbagqabaqcLbmacaWGRbGaey4kaSIaaGymaaaa aaaajuaGbaGaeu4KdC0aaeWaaeaadaWcaaqaaiaaigdaaeaacaWGRb aaaiabgUcaRiaaigdaaiaawIcacaGLPaaadaaeWbqaamaalaaabaGa aiikaiabgkHiTiaaigdacaGGPaWcdaahaaqcfayabeaajugWaiaadQ gaaaqcfa4aaeWaaeaadaqhaaqaaiaadQgaaeaacqaHXoqycqGHsisl caaIXaaaaaGaayjkaiaawMcaaaqaamaabmaabaGaaGymaiabgUcaRi aadQgaaiaawIcacaGLPaaadaahaaqabeaalmaalaaajuaGbaqcLbma caaIXaaajuaGbaqcLbmacaWGRbaaaiabgUcaRiaaigdaaaaaaaqcfa yaaKqzadGaamOAaiabg2da9iaaicdaaKqbagaajugWaiabg6HiLcqc faOaeyyeIuoaaaWaa8qCaeaacaWG4bWaaWbaaeqabaqcLbmacaWGRb aaaKqbakGacwgacaGG4bGaaiiCamaabmaabaGaeyOeI0Iaaiikaiaa dIhacaGGVaGaeq4UdWMaaiykaSWaaWbaaKqbagqabaqcLbmacaWGRb aaaaqcfaOaayjkaiaawMcaamaabmaabaGaaGymaiabgkHiTiGacwga caGG4bGaaiiCamaabmaabaGaeyOeI0IaaiikaiaadIhacaGGVaGaeq 4UdWMaaiykamaaCaaabeqaaKqzadGaam4AaaaaaKqbakaawIcacaGL PaaaaiaawIcacaGLPaaalmaaCaaajuaGbeqaaKqzadGaeqySdeMaey OeI0IaaGymaaaajuaGcqGHciITcaWG4baabaqcLbmacaaIWaaajuaG baqcLbmacaWG4baajuaGcqGHRiI8aaaa@A05F@
Let y= (x/λ) k ;y λ k = x k ;x=λ y 1/k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEai abg2da9iaacIcacaWG4bGaai4laiabeU7aSjaacMcadaahaaqabeaa jugWaiaadUgaaaqcfaOaai4oaiaadMhacqaH7oaBdaahaaqabeaaju gWaiaadUgaaaqcfaOaeyypa0JaamiEamaaCaaabeqaaKqzadGaam4A aaaajuaGcaGG7aGaamiEaiabg2da9iabeU7aSjaadMhalmaaCaaaju aGbeqaaKqzadGaaGymaiaac+cacaWGRbaaaaaa@54DE@
F LBEW (X)= 1 Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 0 x y k exp( y k ) ( 1exp( y k ) ) α1 x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaabaqcLbmacaWGmbGaamOqaiaadweacaWGxbaajuaGbeaacaGG OaGaamiwaiaacMcacqGH9aqpdaWcaaqaaiaaigdaaeaacqqHtoWrda qadaqaamaalaaabaGaaGymaaqaaiaadUgaaaGaey4kaSIaaGymaaGa ayjkaiaawMcaamaaqahabaWaaSaaaeaacaGGOaGaeyOeI0IaaGymai aacMcalmaaCaaajuaGbeqaaKqzadGaamOAaaaajuaGdaqadaqaamaa DaaabaGaamOAaaqaaiabeg7aHjabgkHiTiaaigdaaaaacaGLOaGaay zkaaaabaWaaeWaaeaacaaIXaGaey4kaSIaamOAaaGaayjkaiaawMca amaaCaaabeqaaSWaaSaaaKqbagaajugWaiaaigdaaKqbagaajugWai aadUgaaaGaey4kaSIaaGymaaaaaaaajuaGbaqcLbmacaWGQbGaeyyp a0JaaGimaaqcfayaaKqzadGaeyOhIukajuaGcqGHris5aaaadaWdXb qaaiaadMhadaahaaqabeaajugWaiaadUgaaaqcfaOaciyzaiaacIha caGGWbWaaeWaaeaacqGHsislcaWG5bWcdaahaaqcfayabeaajugWai aadUgaaaaajuaGcaGLOaGaayzkaaWaaeWaaeaacaaIXaGaeyOeI0Ia ciyzaiaacIhacaGGWbWaaeWaaeaacqGHsislcaWG5bWaaWbaaeqaba qcLbmacaWGRbaaaaqcfaOaayjkaiaawMcaaaGaayjkaiaawMcaaSWa aWbaaKqbagqabaqcLbmacqaHXoqycqGHsislcaaIXaaaaKqbakabgk Gi2kaadIhaaeaajugWaiaaicdaaKqbagaajugWaiaadIhaaKqbakab gUIiYdaaaa@91E5@
Let m=y(1+j);y= m 1+j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBai abg2da9iaadMhacaGGOaGaaGymaiabgUcaRiaadQgacaGGPaGaai4o aiaadMhacqGH9aqpdaWcaaqaaiaad2gaaeaacaaIXaGaey4kaSIaam OAaaaaaaa@43B0@
F LBEW (X)= 1 Γ( 1 k +1 ) 0 x m 1/k exp( m )m F LBEW (X)= x 1/k exp( x )[ 1+ 1 kx ] Γ( 1 k +1 )[ 1 1 k + 1 k 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGgbWcdaWgaaqcfayaaKqzadGaamitaiaadkeacaWGfbGaam4vaaqc fayabaGaaiikaiaadIfacaGGPaGaeyypa0ZaaSaaaeaacaaIXaaaba Gaeu4KdC0aaeWaaeaadaWcaaqaaiaaigdaaeaacaWGRbaaaiabgUca RiaaigdaaiaawIcacaGLPaaaaaWaa8qCaeaacaWGTbWaaWbaaeqaba qcLbmacaaIXaGaai4laiaadUgaaaqcfaOaciyzaiaacIhacaGGWbWa aeWaaeaacqGHsislcaWGTbaacaGLOaGaayzkaaGaeyOaIyRaamyBaa qaaKqzadGaaGimaaqcfayaaKqzadGaamiEaaqcfaOaey4kIipaaOqa aKqbakaadAeadaWgaaqaaKqzadGaamitaiaadkeacaWGfbGaam4vaa qcfayabaGaaiikaiaadIfacaGGPaGaeyypa0ZaaSaaaeaacaWG4bWa aWbaaeqabaqcLbmacaaIXaGaai4laiaadUgaaaqcfaOaciyzaiaacI hacaGGWbWaaeWaaeaacqGHsislcaWG4baacaGLOaGaayzkaaWaamWa aeaacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaam4AaiaadIhaaa aacaGLBbGaayzxaaaabaGaeu4KdC0aaeWaaeaadaWcaaqaaiaaigda aeaacaWGRbaaaiabgUcaRiaaigdaaiaawIcacaGLPaaadaWadaqaai aaigdacqGHsisldaWcaaqaaiaaigdaaeaacaWGRbaaaiabgUcaRmaa laaabaGaaGymaaqaaiaadUgadaahaaqabeaajugWaiaaikdaaaaaaa qcfaOaay5waiaaw2faaaaaaaaa@8910@ ………………… (7)

So, the reliability function of LBEW is,

R(x)=1 F LBEW (x)=1 x 1/k exp( x )[ 1+ 1 kx ] Γ( 1 k +1 )[ 1 1 k + 1 k 2 ] R(x)= Γ( 1 k +1 )[ 1 1 k + 1 k 2 ][ 1+ 1 kx ] x 1/k exp( x ) Γ( 1 k +1 )[ 1 1 k + 1 k 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGsbGaaiikaiaadIhacaGGPaGaeyypa0JaaGymaiabgkHiTiaadAea lmaaBaaajuaGbaqcLbmacaWGmbGaamOqaiaadweacaWGxbaajuaGbe aacaGGOaGaamiEaiaacMcacqGH9aqpcaaIXaGaeyOeI0YaaSaaaeaa caWG4bWaaWbaaeqabaqcLbmacaaIXaGaai4laiaadUgaaaqcfaOaci yzaiaacIhacaGGWbWaaeWaaeaacqGHsislcaWG4baacaGLOaGaayzk aaWaamWaaeaacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaam4Aai aadIhaaaaacaGLBbGaayzxaaaabaGaeu4KdC0aaeWaaeaadaWcaaqa aiaaigdaaeaacaWGRbaaaiabgUcaRiaaigdaaiaawIcacaGLPaaada WadaqaaiaaigdacqGHsisldaWcaaqaaiaaigdaaeaacaWGRbaaaiab gUcaRmaalaaabaGaaGymaaqaaiaadUgadaahaaqabeaajugWaiaaik daaaaaaaqcfaOaay5waiaaw2faaaaaaOqaaKqbakaadkfacaGGOaGa amiEaiaacMcacqGH9aqpdaWcaaqaaiabfo5ahnaabmaabaWaaSaaae aacaaIXaaabaGaam4AaaaacqGHRaWkcaaIXaaacaGLOaGaayzkaaWa amWaaeaacaaIXaGaeyOeI0YaaSaaaeaacaaIXaaabaGaam4Aaaaacq GHRaWkdaWcaaqaaiaaigdaaeaacaWGRbWaaWbaaeqabaqcLbmacaaI YaaaaaaaaKqbakaawUfacaGLDbaacqGHsisldaWadaqaaiaaigdacq GHRaWkdaWcaaqaaiaaigdaaeaacaWGRbGaamiEaaaaaiaawUfacaGL DbaacaWG4bWcdaahaaqcfayabeaajugWaiaaigdacaGGVaGaam4Aaa aajuaGciGGLbGaaiiEaiaacchadaqadaqaaiabgkHiTiaadIhaaiaa wIcacaGLPaaaaeaacqqHtoWrdaqadaqaamaalaaabaGaaGymaaqaai aadUgaaaGaey4kaSIaaGymaaGaayjkaiaawMcaamaadmaabaGaaGym aiabgkHiTmaalaaabaGaaGymaaqaaiaadUgaaaGaey4kaSYaaSaaae aacaaIXaaabaGaam4AamaaCaaabeqaaKqzadGaaGOmaaaaaaaajuaG caGLBbGaayzxaaaaaaaaaa@A475@ ……………. (8)

And the hazard function is,

h(x)= f(x) R(x) = k x k λ k+1 exp( (x/λ) k ) ( 1exp( (x/λ) k ) ) α1 [ 1 1 k + 1 k 2 ] { Γ( 1 k +1 )[ 1 1 k + 1 k 2 ][ 1+ 1 kx ] x 1/k exp( x ) } j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAai aacIcacaWG4bGaaiykaiabg2da9maalaaabaGaamOzaiaacIcacaWG 4bGaaiykaaqaaiaadkfacaGGOaGaamiEaiaacMcaaaGaeyypa0ZaaS aaaeaadaWcaaqaaiaadUgacaWG4bWcdaahaaqcfayabeaajugWaiaa dUgaaaaajuaGbaGaeq4UdW2aaWbaaeqajuaibaGaam4AaiabgUcaRi aaigdaaaaaaKqbakGacwgacaGG4bGaaiiCamaabmaabaGaeyOeI0Ia aiikaiaadIhacaGGVaGaeq4UdWMaaiykamaaCaaabeqcfasaaiaadU gaaaaajuaGcaGLOaGaayzkaaWaaeWaaeaacaaIXaGaeyOeI0Iaciyz aiaacIhacaGGWbWaaeWaaeaacqGHsislcaGGOaGaamiEaiaac+cacq aH7oaBcaGGPaWaaWbaaeqajuaibaGaam4AaaaaaKqbakaawIcacaGL PaaaaiaawIcacaGLPaaadaahaaqcfasabeaacqaHXoqycqGHsislca aIXaaaaKqbaoaadmaabaGaaGymaiabgkHiTmaalaaabaGaaGymaaqa aiaadUgaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaam4AamaaCaaabe qcfasaaiaaikdaaaaaaaqcfaOaay5waiaaw2faaaqaamaacmaabaGa eu4KdC0aaeWaaeaadaWcaaqaaiaaigdaaeaacaWGRbaaaiabgUcaRi aaigdaaiaawIcacaGLPaaadaWadaqaaiaaigdacqGHsisldaWcaaqa aiaaigdaaeaacaWGRbaaaiabgUcaRmaalaaabaGaaGymaaqaaiaadU gadaahaaqabKqbGeaacaaIYaaaaaaaaKqbakaawUfacaGLDbaacqGH sisldaWadaqaaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGRb GaamiEaaaaaiaawUfacaGLDbaacaWG4bWaaWbaaeqajuaibaGaaGym aiaac+cacaWGRbaaaKqbakGacwgacaGG4bGaaiiCamaabmaabaGaey OeI0IaamiEaaGaayjkaiaawMcaaaGaay5Eaiaaw2haamaaqahabaWa aSaaaeaacaGGOaGaeyOeI0IaaGymaiaacMcadaahaaqabKqbGeaaca WGQbaaaKqbaoaabmaabaWaa0baaeaacaWGQbaabaGaeqySdeMaeyOe I0IaaGymaaaaaiaawIcacaGLPaaaaeaadaqadaqaaiaaigdacqGHRa WkcaWGQbaacaGLOaGaayzkaaWaaWbaaeqajuaibaqcfa4aaSaaaKqb GeaacaaIXaaabaGaam4AaaaacqGHRaWkcaaIXaaaaaaaaeaacaWGQb Gaeyypa0JaaGimaaqaaiabg6HiLcqcfaOaeyyeIuoaaaaaaa@B34F@ ……. (9)

The moments

The rth raw moment of the LBEW random variable X is

E( X r )= 1 Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 0 x r αk x k λ k+1 exp( (x/λ) k ) ( 1exp( (x/λ) k ) ) α1 x E( X r )= 1 λ r Γ( 1 k +1 ) Γ( r+1 k +1 ) j=0 (1+j) (1/k)+1 ( 1+j ) ((r+1)/k)+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGfbGaaiikaiaadIfadaahaaqabKqbGeaacaWGYbaaaKqbakaacMca cqGH9aqpdaWcaaqaaiaaigdaaeaacqqHtoWrdaqadaqaamaalaaaba GaaGymaaqaaiaadUgaaaGaey4kaSIaaGymaaGaayjkaiaawMcaamaa qahabaWaaSaaaeaacaGGOaGaeyOeI0IaaGymaiaacMcadaahaaqcfa sabeaacaWGQbaaaKqbaoaabmaabaWaa0baaKqbGeaacaWGQbaajuaG baGaeqySdeMaeyOeI0IaaGymaaaaaiaawIcacaGLPaaaaeaadaqada qcfasaaiaaigdacqGHRaWkcaWGQbaacaGLOaGaayzkaaqcfa4aaWba aeqabaWaaSaaaeaacaaIXaaabaGaam4AaaaacqGHRaWkcaaIXaaaaa aaaKqbGeaacaWGQbGaeyypa0JaaGimaaqaaiabg6HiLcqcfaOaeyye IuoaaaWaa8qCaeaacaWG4bWaaWbaaeqajuaibaGaamOCaaaajuaGda Wcaaqaaiabeg7aHjaadUgacaWG4bWaaWbaaeqajuaibaGaam4Aaaaa aKqbagaacqaH7oaBdaahaaqabKqbGeaacaWGRbGaey4kaSIaaGymaa aaaaqcfaOaciyzaiaacIhacaGGWbWaaeWaaeaacqGHsislcaGGOaGa amiEaiaac+cacqaH7oaBcaGGPaWaaWbaaeqajuaibaGaam4AaaaaaK qbakaawIcacaGLPaaadaqadaqaaiaaigdacqGHsislciGGLbGaaiiE aiaacchadaqadaqaaiabgkHiTiaacIcacaWG4bGaai4laiabeU7aSj aacMcadaahaaqabKqbGeaacaWGRbaaaaqcfaOaayjkaiaawMcaaaGa ayjkaiaawMcaamaaCaaabeqcfasaaiabeg7aHjabgkHiTiaaigdaaa qcfaOaeyOaIyRaamiEaaqcfasaaiaaicdaaeaacqGHEisPaKqbakab gUIiYdaakeaajuaGcaWGfbGaaiikaiaadIfadaahaaqcfasabeaaca WGYbaaaKqbakaacMcacqGH9aqpdaWcaaqaaiaaigdaaeaacqaH7oaB daahaaqabKqbGeaacaWGYbaaaKqbakabfo5ahnaabmaabaWaaSaaae aacaaIXaaabaGaam4AaaaacqGHRaWkcaaIXaaacaGLOaGaayzkaaaa aiabfo5ahnaabmaabaWaaSaaaeaacaWGYbGaey4kaSIaaGymaaqaai aadUgaaaGaey4kaSIaaGymaaGaayjkaiaawMcaamaaqahabaWaaSaa aeaacaGGOaGaaGymaiabgUcaRiaadQgacaGGPaWaaWbaaeqajuaiba GaaiikaiaaigdacaGGVaGaam4AaiaacMcacqGHRaWkcaaIXaaaaaqc fayaamaabmaabaGaaGymaiabgUcaRiaadQgaaiaawIcacaGLPaaada ahaaqabeaacaGGOaGaaiikaiaadkhacqGHRaWkcaaIXaGaaiykaiaa c+cacaWGRbGaaiykaiabgUcaRiaaigdaaaaaaaqaaiaadQgacqGH9a qpcaaIWaaajuaibaGaeyOhIukajuaGcqGHris5aaaaaa@C966@ …………. (10)

at r = 1, the first moment of LBEW is

E( X r )= 1 Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 0 x r αk x k λ k+1 exp( (x/λ) k ) ( 1exp( (x/λ) k ) ) α1 x E( X r )= 1 λ r Γ( 1 k +1 ) Γ( r+1 k +1 ) j=0 (1+j) (1/k)+1 ( 1+j ) ((r+1)/k)+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGfbGaaiikaiaadIfadaahaaqabKqbGeaacaWGYbaaaKqbakaacMca cqGH9aqpdaWcaaqaaiaaigdaaeaacqqHtoWrdaqadaqaamaalaaaba GaaGymaaqaaiaadUgaaaGaey4kaSIaaGymaaGaayjkaiaawMcaamaa qahabaWaaSaaaeaacaGGOaGaeyOeI0IaaGymaiaacMcadaahaaqcfa sabeaacaWGQbaaaKqbaoaabmaabaWaa0baaKqbGeaacaWGQbaajuaG baGaeqySdeMaeyOeI0IaaGymaaaaaiaawIcacaGLPaaaaeaadaqada qcfasaaiaaigdacqGHRaWkcaWGQbaacaGLOaGaayzkaaqcfa4aaWba aeqabaWaaSaaaeaacaaIXaaabaGaam4AaaaacqGHRaWkcaaIXaaaaa aaaKqbGeaacaWGQbGaeyypa0JaaGimaaqaaiabg6HiLcqcfaOaeyye IuoaaaWaa8qCaeaacaWG4bWaaWbaaeqajuaibaGaamOCaaaajuaGda Wcaaqaaiabeg7aHjaadUgacaWG4bWaaWbaaeqajuaibaGaam4Aaaaa aKqbagaacqaH7oaBdaahaaqabKqbGeaacaWGRbGaey4kaSIaaGymaa aaaaqcfaOaciyzaiaacIhacaGGWbWaaeWaaeaacqGHsislcaGGOaGa amiEaiaac+cacqaH7oaBcaGGPaWaaWbaaeqajuaibaGaam4AaaaaaK qbakaawIcacaGLPaaadaqadaqaaiaaigdacqGHsislciGGLbGaaiiE aiaacchadaqadaqaaiabgkHiTiaacIcacaWG4bGaai4laiabeU7aSj aacMcadaahaaqabKqbGeaacaWGRbaaaaqcfaOaayjkaiaawMcaaaGa ayjkaiaawMcaamaaCaaabeqcfasaaiabeg7aHjabgkHiTiaaigdaaa qcfaOaeyOaIyRaamiEaaqcfasaaiaaicdaaeaacqGHEisPaKqbakab gUIiYdaakeaajuaGcaWGfbGaaiikaiaadIfadaahaaqcfasabeaaca WGYbaaaKqbakaacMcacqGH9aqpdaWcaaqaaiaaigdaaeaacqaH7oaB daahaaqabKqbGeaacaWGYbaaaKqbakabfo5ahnaabmaabaWaaSaaae aacaaIXaaabaGaam4AaaaacqGHRaWkcaaIXaaacaGLOaGaayzkaaaa aiabfo5ahnaabmaabaWaaSaaaeaacaWGYbGaey4kaSIaaGymaaqaai aadUgaaaGaey4kaSIaaGymaaGaayjkaiaawMcaamaaqahabaWaaSaa aeaacaGGOaGaaGymaiabgUcaRiaadQgacaGGPaWaaWbaaeqajuaiba GaaiikaiaaigdacaGGVaGaam4AaiaacMcacqGHRaWkcaaIXaaaaaqc fayaamaabmaabaGaaGymaiabgUcaRiaadQgaaiaawIcacaGLPaaada ahaaqabeaacaGGOaGaaiikaiaadkhacqGHRaWkcaaIXaGaaiykaiaa c+cacaWGRbGaaiykaiabgUcaRiaaigdaaaaaaaqaaiaadQgacqGH9a qpcaaIWaaajuaibaGaeyOhIukajuaGcqGHris5aaaaaa@C966@ …………. (11)

And the variance is
                                                                                         Var(X)= 1 λ 2 Γ( 1 k +1 ) Γ( 3 k +1 ) j=0 (1+j) (1/k) [ 1 λΓ( 1 k +1 ) Γ( 2 k +1 ) j=0 (1+j) (1/k) ] 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvai aadggacaWGYbGaaiikaiaadIfacaGGPaGaeyypa0ZaaSaaaeaacaaI XaaabaGaeq4UdW2aaWbaaeqajuaibaGaaGOmaaaajuaGcqqHtoWrda qadaqaamaalaaabaGaaGymaaqaaiaadUgaaaGaey4kaSIaaGymaaGa ayjkaiaawMcaaaaacqqHtoWrdaqadaqaamaalaaabaGaaG4maaqaai aadUgaaaGaey4kaSIaaGymaaGaayjkaiaawMcaamaaqahabaGaaiik aiaaigdacqGHRaWkcaWGQbGaaiykamaaCaaabeqcfasaaiabgkHiTi aacIcacaaIXaGaai4laiaadUgacaGGPaaaaaqaaiaadQgacqGH9aqp caaIWaaabaGaeyOhIukajuaGcqGHris5aiabgkHiTmaadmaabaWaaS aaaeaacaaIXaaabaGaeq4UdWMaeu4KdC0aaeWaaeaadaWcaaqaaiaa igdaaeaacaWGRbaaaiabgUcaRiaaigdaaiaawIcacaGLPaaaaaGaeu 4KdC0aaeWaaeaadaWcaaqaaiaaikdaaeaacaWGRbaaaiabgUcaRiaa igdaaiaawIcacaGLPaaadaaeWbqaaiaacIcacaaIXaGaey4kaSIaam OAaiaacMcadaahaaqabKqbGeaacqGHsislcaGGOaGaaGymaiaac+ca caWGRbGaaiykaaaaaeaacaWGQbGaeyypa0JaaGimaaqaaiabg6HiLc qcfaOaeyyeIuoaaiaawUfacaGLDbaadaahaaqabKqbGeaacaaIYaaa aaaa@7F84@ …….. (12)

The skewness and kurtosis of LBEW;

k 3 = αλΓ( 3 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 3 k +1 [ 6 α 2 λΓ( 4 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 4 k +1 ][ α 2 λΓ( 2 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 2 k +1 ]+8 α 2 λ 2 Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 [ Var(X) ] 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aam aaBaaajuaibaGaaG4maaqcfayabaGaeyypa0ZaaSaaaeaacqaHXoqy cqaH7oaBcqqHtoWrdaqadaqaamaalaaabaGaaG4maaqaaiaadUgaaa Gaey4kaSIaaGymaaGaayjkaiaawMcaamaaqahabaWaaSaaaeaacaGG OaGaeyOeI0IaaGymaiaacMcadaahaaqabKqbGeaacaWGQbaaaKqbao aabmaajqwba+FaaKqbaoaaDaaajqwba+FaaiaadQgaaeaacqaHXoqy cqGHsislcaaIXaaaaaGaayjkaiaawMcaaaqcfayaamaabmaajuaiba GaaGymaiabgUcaRiaadQgaaiaawIcacaGLPaaajuaGdaahaaqabeaa daWcaaqaaiaaiodaaeaacaWGRbaaaiabgUcaRiaaigdaaaaaaiabgk HiTmaadmaabaGaaGOnaiabeg7aHnaaCaaajuaibeqaaiaaikdaaaqc faOaeq4UdWMaeu4KdC0aaeWaaeaadaWcaaqaaiaaisdaaeaacaWGRb aaaiabgUcaRiaaigdaaiaawIcacaGLPaaadaaeWbqaamaalaaabaGa aiikaiabgkHiTiaaigdacaGGPaWaaWbaaeqajuaibaGaamOAaaaaju aGdaqadaqaamaaDaaabaGaamOAaaqaaiabeg7aHjabgkHiTiaaigda aaaacaGLOaGaayzkaaaabaWaaeWaaeaacaaIXaGaey4kaSIaamOAaa GaayjkaiaawMcaamaaCaaabeqaamaalaaabaGaaGinaaqaaiaadUga aaGaey4kaSIaaGymaaaaaaaajuaibaGaamOAaiabg2da9iaaicdaae aacqGHEisPaKqbakabggHiLdaacaGLBbGaayzxaaWaamWaaeaacqaH XoqydaahaaqabKqbGeaacaaIYaaaaKqbakabeU7aSjabfo5ahnaabm aabaWaaSaaaeaacaaIYaaabaGaam4AaaaacqGHRaWkcaaIXaaacaGL OaGaayzkaaWaaabCaeaadaWcaaqaaiaacIcacqGHsislcaaIXaGaai ykamaaCaaabeqcfasaaiaadQgaaaqcfa4aaeWaaeaadaqhaaqaaiaa dQgaaeaacqaHXoqycqGHsislcaaIXaaaaaGaayjkaiaawMcaaaqaam aabmaabaGaaGymaiabgUcaRiaadQgaaiaawIcacaGLPaaadaahaaqa beaadaWcaaqaaiaaikdaaeaacaWGRbaaaiabgUcaRiaaigdaaaaaaa qcfasaaiaadQgacqGH9aqpcaaIWaaabaGaeyOhIukajuaGcqGHris5 aaGaay5waiaaw2faaiabgUcaRiaaiIdacqaHXoqydaahaaqabKqbGe aacaaIYaaaaKqbakabeU7aSnaaCaaabeqcfasaaiaaikdaaaqcfaOa eu4KdC0aaeWaaeaadaWcaaqaaiaaigdaaeaacaWGRbaaaiabgUcaRi aaigdaaiaawIcacaGLPaaadaaeWbqaamaalaaabaGaaiikaiabgkHi TiaaigdacaGGPaWaaWbaaeqajuaibaGaamOAaaaajuaGdaqadaqaam aaDaaabaGaamOAaaqaaiabeg7aHjabgkHiTiaaigdaaaaacaGLOaGa ayzkaaaabaWaaeWaaeaacaaIXaGaey4kaSIaamOAaaGaayjkaiaawM caamaaCaaabeqaamaalaaabaGaaGymaaqaaiaadUgaaaGaey4kaSIa aGymaaaaaaaajuaibaGaamOAaiabg2da9iaaicdaaeaacqGHEisPaK qbakabggHiLdaajuaibaGaamOAaiabg2da9iaaicdaaeaacqGHEisP aKqbakabggHiLdaabaWaamWaaeaacaWGwbGaamyyaiaadkhacaGGOa GaamiwaiaacMcaaiaawUfacaGLDbaadaahaaqabKqbGeaacaaIZaGa ai4laiaaikdaaaaaaaaa@E2A3@ …..(13)

k 4 = α λ 2 Γ( 4 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 4 k +1 [ 8 α 2 λ 2 Γ( 3 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 3 k +1 ][ α 2 λΓ( 4 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 4 k +1 ]+[ 6 α 2 λ 2 Γ( 2 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 2 k +1 ] [ Var(X) ] 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aam aaBaaajuaibaGaaGinaaqcfayabaGaeyypa0ZaaSaaaeaacqaHXoqy cqaH7oaBdaahaaqabKqbGeaacaaIYaaaaKqbakabfo5ahnaabmaaba WaaSaaaeaacaaI0aaabaGaam4AaaaacqGHRaWkcaaIXaaacaGLOaGa ayzkaaWaaabCaeaadaWcaaqaaiaacIcacqGHsislcaaIXaGaaiykam aaCaaabeqcfasaaiaadQgaaaqcfa4aaeWaaeaadaqhaaqaaiaadQga aeaacqaHXoqycqGHsislcaaIXaaaaaGaayjkaiaawMcaaaqaamaabm aabaGaaGymaiabgUcaRiaadQgaaiaawIcacaGLPaaadaahaaqabeaa daWcaaqaaiaaisdaaeaacaWGRbaaaiabgUcaRiaaigdaaaaaaiabgk HiTmaadmaabaGaaGioaiabeg7aHnaaCaaabeqcfasaaiaaikdaaaqc faOaeq4UdW2aaWbaaeqajuaibaGaaGOmaaaajuaGcqqHtoWrdaqada qaamaalaaabaGaaG4maaqaaiaadUgaaaGaey4kaSIaaGymaaGaayjk aiaawMcaamaaqahabaWaaSaaaeaacaGGOaGaeyOeI0IaaGymaiaacM cadaahaaqabKqbGeaacaWGQbaaaKqbaoaabmaabaWaa0baaeaacaWG QbaabaGaeqySdeMaeyOeI0IaaGymaaaaaiaawIcacaGLPaaaaeaada qadaqaaiaaigdacqGHRaWkcaWGQbaacaGLOaGaayzkaaWaaWbaaeqa baWaaSaaaeaacaaIZaaabaGaam4AaaaacqGHRaWkcaaIXaaaaaaaaK qbGeaacaWGQbGaeyypa0JaaGimaaqaaiabg6HiLcqcfaOaeyyeIuoa aiaawUfacaGLDbaadaWadaqaaiabeg7aHnaaCaaajuaibeqaaiaaik daaaqcfaOaeq4UdWMaeu4KdC0aaeWaaeaadaWcaaqaaiaaisdaaeaa caWGRbaaaiabgUcaRiaaigdaaiaawIcacaGLPaaadaaeWbqaamaala aabaGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaeqajuaibaGaamOA aaaajuaGdaqadaqaamaaDaaabaGaamOAaaqaaiabeg7aHjabgkHiTi aaigdaaaaacaGLOaGaayzkaaaabaWaaeWaaeaacaaIXaGaey4kaSIa amOAaaGaayjkaiaawMcaamaaCaaabeqaamaalaaabaGaaGinaaqaai aadUgaaaGaey4kaSIaaGymaaaaaaaajuaibaGaamOAaiabg2da9iaa icdaaeaacqGHEisPaKqbakabggHiLdaacaGLBbGaayzxaaGaey4kaS YaamWaaeaacaaI2aGaeqySde2aaWbaaKqbGeqabaGaaGOmaaaajuaG cqaH7oaBdaahaaqabKqbGeaacaaIYaaaaKqbakabfo5ahnaabmaaba WaaSaaaeaacaaIYaaabaGaam4AaaaacqGHRaWkcaaIXaaacaGLOaGa ayzkaaWaaabCaeaadaWcaaqaaiaacIcacqGHsislcaaIXaGaaiykam aaCaaabeqcfasaaiaadQgaaaqcfa4aaeWaaeaadaqhaaqaaiaadQga aeaacqaHXoqycqGHsislcaaIXaaaaaGaayjkaiaawMcaaaqaamaabm aabaGaaGymaiabgUcaRiaadQgaaiaawIcacaGLPaaadaahaaqabeaa daWcaaqaaiaaikdaaeaacaWGRbaaaiabgUcaRiaaigdaaaaaaaqcfa saaiaadQgacqGH9aqpcaaIWaaabaGaeyOhIukajuaGcqGHris5aaGa ay5waiaaw2faaaqcfasaaiaadQgacqGH9aqpcaaIWaaabaGaeyOhIu kajuaGcqGHris5aaqaamaadmaabaGaamOvaiaadggacaWGYbGaaiik aiaadIfacaGGPaaacaGLBbGaayzxaaWaaWbaaeqajuaibaGaaGOmaa aaaaaaaa@E0A6@           …..(14)

From equation (7), (8), (9), (10), (11), (12), (13), (14), we established all the properties of LBEW distribution and also that of ABEW was obtained which can be fetch in the body of the work.

Maximum likelihood approach

Harter and Moore (1965) were the earliest statisticians to use the maximum likelihood procedure because of its desirable characteristics.
The three distributions in the study (EW, LBEW and ABEW) are solved iteratively by computer algorithm to obtain the maximum likelihood estimates of the parameters α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ , k and λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ .

MLE of EW

Let Xbe a random sample of size n from the EW distribution given by equation (1). Then the log likelihood function comes out to be

L(α,λ,k)=nlnα+nlnk+nklnλ+(k1) ln x i + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai aacIcacqaHXoqycaGGSaGaeq4UdWMaaiilaiaadUgacaGGPaGaeyyp a0JaamOBaiGacYgacaGGUbGaeqySdeMaey4kaSIaamOBaiGacYgaca GGUbGaam4AaiabgUcaRiaad6gacaWGRbGaciiBaiaac6gacqaH7oaB cqGHRaWkcaGGOaGaam4AaiabgkHiTiaaigdacaGGPaWaaabqaeaaci GGSbGaaiOBaiaadIhadaWgaaqcfasaaiaadMgaaeqaaaqcfayabeqa cqGHris5aiabgUcaRaaa@5B30@
(α1) ln[ 1exp{ ( x i /λ) k } ] ( x i /λ) k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai abeg7aHjabgkHiTiaaigdacaGGPaWaaabqaeaaciGGSbGaaiOBamaa dmaabaGaaGymaiabgkHiTiGacwgacaGG4bGaaiiCaiaacUhacqGHsi slcaGGOaGaamiEamaaBaaabaGaamyAaaqabaGaai4laiabeU7aSjaa cMcadaahaaqabKqbGeaacaWGRbaaaKqbakaac2haaiaawUfacaGLDb aaaeqabeGaeyyeIuoacqGHsisldaaeabqaaiaacIcacaWG4bWaaSba aeaacaWGPbaabeaacaGGVaGaeq4UdWMaaiykamaaCaaabeqcfasaai aadUgaaaaajuaGbeqabiabggHiLdaaaa@5A83@     ….. (15)               
Therefore the MLEs of α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ , λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ , k which maximize (15) must satisfy the normal equations given by
Derivative w.r.t α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@
α L(α,λ,k)= n α + ln[ 1exp{ ( x i /λ) k } ] =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITaeaacqGHciITcqaHXoqyaaGaamitaiaacIcacqaHXoqy caGGSaGaeq4UdWMaaiilaiaadUgacaGGPaGaeyypa0ZaaSaaaeaaca WGUbaabaGaeqySdegaaiabgUcaRmaaqaeabaGaciiBaiaac6gadaWa daqaaiaaigdacqGHsislciGGLbGaaiiEaiaacchacaGG7bGaeyOeI0 IaaiikaiaadIhadaWgaaqaaiaadMgaaeqaaiaac+cacqaH7oaBcaGG PaWaaWbaaKqbGeqabaGaam4AaaaajuaGcaGG9baacaGLBbGaayzxaa aabeqabiabggHiLdGaeyypa0JaaGimaaaa@5DF2@

We obtain the MLE of

λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ as
α Λ = n ln[ 1exp{ ( x i /λ) k } ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbiae aacqaHXoqyaeqajuaibaGaeu4MdWeaaKqbakabg2da9iabgkHiTmaa laaabaGaamOBaaqaamaaqaeabaGaciiBaiaac6gadaWadaqaaiaaig dacqGHsislciGGLbGaaiiEaiaacchacaGG7bGaeyOeI0Iaaiikaiaa dIhadaWgaaqcfasaaiaadMgaaKqbagqaaiaac+cacqaH7oaBcaGGPa WaaWbaaKqbGeqabaGaam4AaaaajuaGcaGG9baacaGLBbGaayzxaaaa beqabiabggHiLdaaaaaa@5320@
      ….. (16)

Derivative w.r.t λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@

λ L(α,λ,k)= nk λ +(α1)k λ k1 exp{ ( x i /λ) k } 1exp{ ( x i /λ) k } x i k λ k1 x i k =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITaeaacqGHciITcqaH7oaBaaGaamitaiaacIcacqaHXoqy caGGSaGaeq4UdWMaaiilaiaadUgacaGGPaGaeyypa0ZaaSaaaeaaca WGUbGaam4AaaqaaiabeU7aSbaacqGHRaWkcaGGOaGaeqySdeMaeyOe I0IaaGymaiaacMcacaWGRbGaeq4UdW2aaWbaaeqajuaibaGaam4Aai abgkHiTiaaigdaaaqcfa4aaabqaeaadaWcaaqaaiGacwgacaGG4bGa aiiCaiaacUhacqGHsislcaGGOaGaamiEamaaBaaajuaibaGaamyAaa qcfayabaGaai4laiabeU7aSjaacMcadaahaaqabKqbGeaacaWGRbaa aKqbakaac2haaeaacaaIXaGaeyOeI0IaciyzaiaacIhacaGGWbGaai 4EaiabgkHiTiaacIcacaWG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaa caGGVaGaeq4UdWMaaiykamaaCaaabeqaaiaadUgaaaGaaiyFaaaaae qabeGaeyyeIuoacaWG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaacqGH sislcaWGRbGaeq4UdW2aaWbaaeqajuaibaGaam4AaiabgkHiTiaaig daaaqcfa4aaabqaeaacaWG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaa daahaaqabKqbGeaacaWGRbaaaaqcfayabeqacqGHris5aiabg2da9i aaicdaaaa@847F@

Multiplying the above equation by

λ k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH7oaBaeaacaWGRbaaaaaa@3938@ we get
n+ λ k [ (α1) exp{ ( x i /λ) k } 1exp{ ( x i /λ) k } x i k x i k ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBai abgUcaRiabeU7aSnaaCaaabeqcfasaaiaadUgaaaqcfa4aamWaaeaa caGGOaGaeqySdeMaeyOeI0IaaGymaiaacMcadaaeabqaamaalaaaba GaciyzaiaacIhacaGGWbGaai4EaiabgkHiTiaacIcacaWG4bWaaSba aKqbGeaacaWGPbaabeaajuaGcaGGVaGaeq4UdWMaaiykamaaCaaabe qcfasaaiaadUgaaaqcfaOaaiyFaaqaaiaaigdacqGHsislciGGLbGa aiiEaiaacchacaGG7bGaeyOeI0IaaiikaiaadIhadaWgaaqcfasaai aadMgaaeqaaKqbakaac+cacqaH7oaBcaGGPaWaaWbaaeqajuaibaGa am4AaaaajuaGcaGG9baaaaqabeqacqGHris5aiaadIhadaWgaaqcfa saaiaadMgaaeqaaKqbaoaaCaaajuaibeqaaiaadUgaaaqcfaOaeyOe I0YaaabqaeaacaWG4bWaaSbaaKqbGeaacaWGPbaabeaajuaGdaahaa qcfasabeaacaWGRbaaaaqcfayabeqacqGHris5aaGaay5waiaaw2fa aiabg2da9iaaicdaaaa@7032@ …... (17)
Derivative w.r.t k

k L(α,λ,k)= n k +nlnλ+ ln x i +(α1) λ k exp{ ( x i /λ) k } 1exp{ ( x i /λ) k } x i k ln( x i /λ) λ k1 x i k ln( x i /λ)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITaeaacqGHciITcaWGRbaaaiaadYeacaGGOaGaeqySdeMa aiilaiabeU7aSjaacYcacaWGRbGaaiykaiabg2da9maalaaabaGaam OBaaqaaiaadUgaaaGaey4kaSIaamOBaiGacYgacaGGUbGaeq4UdWMa ey4kaSYaaabqaeaaciGGSbGaaiOBaaqabeqacqGHris5aiaadIhada WgaaqcfasaaiaadMgaaKqbagqaaiabgUcaRiaacIcacqaHXoqycqGH sislcaaIXaGaaiykaiabeU7aSnaaCaaabeqcfasaaiaadUgaaaqcfa 4aaabqaeaadaWcaaqaaiGacwgacaGG4bGaaiiCaiaacUhacqGHsisl caGGOaGaamiEamaaBaaajuaibaGaamyAaaqcfayabaGaai4laiabeU 7aSjaacMcadaahaaqabKqbGeaacaWGRbaaaKqbakaac2haaeaacaaI XaGaeyOeI0IaciyzaiaacIhacaGGWbGaai4EaiabgkHiTiaacIcaca WG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaacaGGVaGaeq4UdWMaaiyk amaaCaaabeqcfasaaiaadUgaaaqcfaOaaiyFaaaaaeqabeGaeyyeIu oacaWG4bWaaSbaaKqbGeaacaWGPbaabeaajuaGdaahaaqcfasabeaa caWGRbaaaKqbakGacYgacaGGUbGaaiikaiaadIhadaWgaaqcfasaai aadMgaaKqbagqaaiaac+cacqaH7oaBcaGGPaGaeyOeI0Iaeq4UdW2a aWbaaeqajuaibaGaam4AaiabgkHiTiaaigdaaaqcfa4aaabqaeaaca WG4bWaaSbaaKqbGeaacaWGPbaabeaajuaGdaahaaqcfasabeaacaWG RbaaaaqcfayabeqacqGHris5aiGacYgacaGGUbGaaiikaiaadIhada WgaaqcfauaaiaadMgaaeqaaKqbakaac+cacqaH7oaBcaGGPaGaeyyp a0JaaGimaaaa@9EF8@


then,
n k + ln x i + λ k [ (α1) exp{ ( x i /λ) k } 1exp{ ( x i /λ) k } x i k ln x i x i k ln x i ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGUbaabaGaam4AaaaacqGHRaWkdaaeabqaaiGacYgacaGGUbaa beqabiabggHiLdGaamiEamaaBaaajuaibaGaamyAaaqabaqcfaOaey 4kaSIaeq4UdW2aaWbaaeqajuaibaGaam4AaaaajuaGdaWadaqaaiaa cIcacqaHXoqycqGHsislcaaIXaGaaiykamaaqaeabaWaaSaaaeaaci GGLbGaaiiEaiaacchacaGG7bGaeyOeI0IaaiikaiaadIhadaWgaaqc fasaaiaadMgaaeqaaKqbakaac+cacqaH7oaBcaGGPaWaaWbaaeqaju aibaGaam4AaaaajuaGcaGG9baabaGaaGymaiabgkHiTiGacwgacaGG 4bGaaiiCaiaacUhacqGHsislcaGGOaGaamiEamaaBaaajuaibaGaam yAaaqcfayabaGaai4laiabeU7aSjaacMcadaahaaqabKqbGeaacaWG RbaaaKqbakaac2haaaaabeqabiabggHiLdGaamiEamaaBaaajuaiba GaamyAaaqabaqcfa4aaWbaaKqbGeqabaGaam4AaaaajuaGciGGSbGa aiOBaiaadIhadaWgaaqcfasaaiaadMgaaKqbagqaaiabgkHiTmaaqa eabaGaamiEamaaBaaajuaibaGaamyAaaqabaqcfa4aaWbaaKqbGeqa baGaam4AaaaajuaGciGGSbGaaiOBaiaadIhadaWgaaqcfasaaiaadM gaaKqbagqaaaqabeqacqGHris5aaGaay5waiaaw2faaiabg2da9iaa icdaaaa@8214@      ……… (18)

Using (15) in (17) and (18) we get equations, which are satisfied by the MLEs λ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbiae aacuaH7oaBgaqcaaqabeaaaaaaaa@3886@ and k ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaKqbag aaceWGRbGbaKaaaSqabeaaaaaaaa@37CD@  of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ and k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbaaaa@3794@ , respectively. Because of the complicated form of the likelihood equations, algebraically it is very difficult to prove that the solution to the normal equations give a global maximum or at least a local maximum, though numerical computation during data analysis showed the presence of at least local maximum.

However, the following properties of the log-likelihood function have been algebraically noted:

  1. for given ( λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ , k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbaaaa@3794@ ), log-likelihood is a strictly concave function of α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ . Further, the optimal value of α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ , given by (8), is a concave increasing function of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ , for given k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbaaaa@3794@ ;
  2. for given ( α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ , k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbaaaa@3794@ ), and α1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde MaeyyzImRaaGymaaaa@3AA4@ , log-likelihood is a strictly concave function of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@

MLE of LBEW

Taking the log-likelihood and derivative of the equation (4) to obtain the MLEs of parameters α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ , k and λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@

α L(α,λ,k)= i=1 n (α1)ln [ 1exp{ ( x i /λ) k } ] α2 α { nln[ Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 ] }=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITaeaacqGHciITcqaHXoqyaaGaamitaiaacIcacqaHXoqy caGGSaGaeq4UdWMaaiilaiaadUgacaGGPaGaeyypa0ZaaabCaeaaca GGOaGaeqySdeMaeyOeI0IaaGymaiaacMcaciGGSbGaaiOBaaqcfasa aiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIuoadaWada qaaiaaigdacqGHsislciGGLbGaaiiEaiaacchacaGG7bGaeyOeI0Ia aiikaiaadIhadaWgaaqcfasaaiaadMgaaKqbagqaaiaac+cacqaH7o aBcaGGPaWaaWbaaKqbGeqabaGaam4AaaaajuaGcaGG9baacaGLBbGa ayzxaaWaaWbaaeqajuaibaGaeqySdeMaeyOeI0IaaGOmaaaajuaGcq GHsisldaWcaaqaaiabgkGi2cqaaiabgkGi2kabeg7aHbaadaGadaqa aiaad6gaciGGSbGaaiOBamaadmaabaGaeu4KdC0aaeWaaeaadaWcaa qaaiaaigdaaeaacaWGRbaaaiabgUcaRiaaigdaaiaawIcacaGLPaaa daaeWbqaamaalaaabaGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaae qajuaibaGaamOAaaaajuaGdaqadaqaamaaDaaabaGaamOAaaqaaiab eg7aHjabgkHiTiaaigdaaaaacaGLOaGaayzkaaaabaWaaeWaaeaaca aIXaGaey4kaSIaamOAaaGaayjkaiaawMcaamaaCaaabeqaamaalaaa baGaaGymaaqaaiaadUgaaaGaey4kaSIaaGymaaaaaaaajuaibaGaam OAaiabg2da9iaaicdaaeaacqGHEisPaKqbakabggHiLdaacaGLBbGa ayzxaaaacaGL7bGaayzFaaGaeyypa0JaaGimaaaa@947B@

               ……… (19)

k L(α,λ,k)= n k + i=1 n kln x i k1 x i k nlnλk (x/λ) k i=1 n { (α1)exp{(k x i k / λ k+1 )} 1exp{ ( x i /λ) k } } α1 α { nln[ Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 ] }=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITaeaacqGHciITcaWGRbaaaiaadYeacaGGOaGaeqySdeMa aiilaiabeU7aSjaacYcacaWGRbGaaiykaiabg2da9maalaaabaGaam OBaaqaaiaadUgaaaGaey4kaSYaaabCaeaadaWcaaqaaiaadUgaciGG SbGaaiOBaiaadIhadaWgaaqcfasaaiaadMgaaeqaaKqbaoaaCaaaju aibeqaaiaadUgacqGHsislcaaIXaaaaaqcfayaaiaadIhadaWgaaqc fasaaiaadMgaaKqbagqaamaaCaaajuaibeqaaiaadUgaaaaaaaqaai aadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIuoacqGHsisl caWGUbGaciiBaiaac6gacqaH7oaBcqGHsislcaWGRbGaaiikaiaadI hacaGGVaGaeq4UdWMaaiykamaaCaaabeqcfasaaiaadUgaaaqcfaOa eyOeI0YaaabCaeaadaGadaqaamaalaaabaGaaiikaiabeg7aHjabgk HiTiaaigdacaGGPaGaciyzaiaacIhacaGGWbGaai4EaiabgkHiTiaa cIcacaWGRbGaamiEamaaBaaajuaibaGaamyAaaqabaqcfa4aaWbaaK qbGeqabaGaam4AaaaajuaGcaGGVaGaeq4UdW2aaWbaaeqajuaibaGa am4AaiabgUcaRiaaigdaaaqcfaOaaiykaiaac2haaeaacaaIXaGaey OeI0IaciyzaiaacIhacaGGWbGaai4EaiabgkHiTiaacIcacaWG4bWa aSbaaKqbGeaacaWGPbaabeaajuaGcaGGVaGaeq4UdWMaaiykamaaCa aabeqcfasaaiaadUgaaaqcfaOaaiyFaaaaaiaawUhacaGL9baaaKqb GeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaaKqbakabggHiLdWaaW baaeqajuaibaGaeqySdeMaeyOeI0IaaGymaaaajuaGcqGHsisldaWc aaqaaiabgkGi2cqaaiabgkGi2kabeg7aHbaadaGadaqaaiaad6gaci GGSbGaaiOBamaadmaabaGaeu4KdC0aaeWaaeaadaWcaaqaaiaaigda aeaacaWGRbaaaiabgUcaRiaaigdaaiaawIcacaGLPaaadaaeWbqaam aalaaabaGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaeqajuaibaGa amOAaaaajuaGdaqadaqaamaaDaaabaGaamOAaaqaaiabeg7aHjabgk HiTiaaigdaaaaacaGLOaGaayzkaaaabaWaaeWaaeaacaaIXaGaey4k aSIaamOAaaGaayjkaiaawMcaamaaCaaabeqaamaalaaabaGaaGymaa qaaiaadUgaaaGaey4kaSIaaGymaaaaaaaajuaibaGaamOAaiabg2da 9iaaicdaaeaacqGHEisPaKqbakabggHiLdaacaGLBbGaayzxaaaaca GL7bGaayzFaaGaeyypa0JaaGimaaaa@C9D5@

                                                                                                                                                                                       
.… (20)

λ L(α,λ,k)= n(k+1) λ + k x i k λ k+1 i=1 n { (α1)exp{(k x i k / λ k+1 )} 1exp{ ( x i /λ) k } } α1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITaeaacqGHciITcqaH7oaBaaGaamitaiaacIcacqaHXoqy caGGSaGaeq4UdWMaaiilaiaadUgacaGGPaGaeyypa0JaeyOeI0YaaS aaaeaacaWGUbGaaiikaiaadUgacqGHRaWkcaaIXaGaaiykaaqaaiab eU7aSbaacqGHRaWkdaWcaaqaaiaadUgacaWG4bWaaSbaaKqbGeaaca WGPbaabeaajuaGdaahaaqcfasabeaacaWGRbaaaaqcfayaaiabeU7a SnaaCaaabeqcfasaaiaadUgacqGHRaWkcaaIXaaaaaaajuaGcqGHsi sldaaeWbqaamaacmaabaWaaSaaaeaacaGGOaGaeqySdeMaeyOeI0Ia aGymaiaacMcaciGGLbGaaiiEaiaacchacaGG7bGaeyOeI0Iaaiikai aadUgacaWG4bWaaSbaaKqbGeaacaWGPbaabeaajuaGdaahaaqcfasa beaacaWGRbaaaKqbakaac+cacqaH7oaBdaahaaqabKqbGeaacaWGRb Gaey4kaSIaaGymaaaajuaGcaGGPaGaaiyFaaqaaiaaigdacqGHsisl ciGGLbGaaiiEaiaacchacaGG7bGaeyOeI0IaaiikaiaadIhadaWgaa qcfauaaiaadMgaaeqaaKqbakaac+cacqaH7oaBcaGGPaWaaWbaaeqa juaibaGaam4AaaaajuaGcaGG9baaaaGaay5Eaiaaw2haaaqcfasaai aadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIuoadaahaaqa bKqbGeaacqaHXoqycqGHsislcaaIXaaaaKqbakabg2da9iaaicdaaa a@8DA2@         …. (21)

Equations (19), (20) and (21) are solved iteratively to obtain the maximum likelihood estimates of the parameters α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ , k and λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ .

MLE of ABEW

Taking the log-likelihood and derivative of the equation (6) to obtain the MLEs of parameters α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ , k and λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@

α L(α,λ,k)= i=1 n (α1)ln [ 1exp{ ( x i /λ) k } ] α2 α { nln[ Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 ] }=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITaeaacqGHciITcqaHXoqyaaGaamitaiaacIcacqaHXoqy caGGSaGaeq4UdWMaaiilaiaadUgacaGGPaGaeyypa0ZaaabCaeaaca GGOaGaeqySdeMaeyOeI0IaaGymaiaacMcaciGGSbGaaiOBaaqcfasa aiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIuoadaWada qaaiaaigdacqGHsislciGGLbGaaiiEaiaacchacaGG7bGaeyOeI0Ia aiikaiaadIhadaWgaaqcfasaaiaadMgaaeqaaKqbakaac+cacqaH7o aBcaGGPaWaaWbaaeqajuaibaGaam4AaaaajuaGcaGG9baacaGLBbGa ayzxaaWaaWbaaeqajuaibaGaeqySdeMaeyOeI0IaaGOmaaaajuaGcq GHsisldaWcaaqaaiabgkGi2cqaaiabgkGi2kabeg7aHbaadaGadaqa aiaad6gaciGGSbGaaiOBamaadmaabaGaeu4KdC0aaeWaaeaadaWcaa qaaiaaigdaaeaacaWGRbaaaiabgUcaRiaaigdaaiaawIcacaGLPaaa daaeWbqaamaalaaabaGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaae qajuaibaGaamOAaaaajuaGdaqadaqaamaaDaaabaGaamOAaaqaaiab eg7aHjabgkHiTiaaigdaaaaacaGLOaGaayzkaaaabaWaaeWaaeaaca aIXaGaey4kaSIaamOAaaGaayjkaiaawMcaamaaCaaabeqaamaalaaa baGaaGymaaqaaiaadUgaaaGaey4kaSIaaGymaaaaaaaajuaibaGaam OAaiabg2da9iaaicdaaeaacqGHEisPaKqbakabggHiLdaacaGLBbGa ayzxaaaacaGL7bGaayzFaaGaeyypa0JaaGimaaaa@947B@

…………..(22)

k L(α,λ,k)= n k + i=1 n (k+1)ln x i k x i k+1 nlnλk (x/λ) k i=1 n { (α1)exp{(k x i k / λ k+1 )} 1exp{ ( x i /λ) k } } α1 α { nln[ Γ( 1 k +1 ) j=0 (1) j ( j α1 ) ( 1+j ) 1 k +1 ] }=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITaeaacqGHciITcaWGRbaaaiaadYeacaGGOaGaeqySdeMa aiilaiabeU7aSjaacYcacaWGRbGaaiykaiabg2da9maalaaabaGaam OBaaqaaiaadUgaaaGaey4kaSYaaabCaeaadaWcaaqaaiaacIcacaWG RbGaey4kaSIaaGymaiaacMcaciGGSbGaaiOBaiaadIhadaWgaaqcfa saaiaadMgaaeqaaKqbaoaaCaaajuaibeqaaiaadUgaaaaajuaGbaGa amiEamaaBaaajuaibaGaamyAaaqabaqcfa4aaWbaaKqbGeqabaGaam 4AaiabgUcaRiaaigdaaaaaaaqaaiaadMgacqGH9aqpcaaIXaaabaGa amOBaaqcfaOaeyyeIuoacqGHsislcaWGUbGaciiBaiaac6gacqaH7o aBcqGHsislcaWGRbGaaiikaiaadIhacaGGVaGaeq4UdWMaaiykamaa CaaabeqcfasaaiaadUgaaaqcfaOaeyOeI0YaaabCaeaadaGadaqaam aalaaabaGaaiikaiabeg7aHjabgkHiTiaaigdacaGGPaGaciyzaiaa cIhacaGGWbGaai4EaiabgkHiTiaacIcacaWGRbGaamiEamaaBaaaju aibaGaamyAaaqabaqcfa4aaWbaaKqbGeqabaGaam4AaaaajuaGcaGG VaGaeq4UdW2aaWbaaeqajuaibaGaam4AaiabgUcaRiaaigdaaaqcfa Oaaiykaiaac2haaeaacaaIXaGaeyOeI0IaciyzaiaacIhacaGGWbGa ai4EaiabgkHiTiaacIcacaWG4bWaaSbaaKqbGeaacaWGPbaabeaaju aGcaGGVaGaeq4UdWMaaiykamaaCaaabeqcfasaaiaadUgaaaqcfaOa aiyFaaaaaiaawUhacaGL9baaaKqbGeaacaWGPbGaeyypa0JaaGymaa qaaiaad6gaaKqbakabggHiLdWaaWbaaeqajuaibaGaeqySdeMaeyOe I0IaaGymaaaajuaGcqGHsisldaWcaaqaaiabgkGi2cqaaiabgkGi2k abeg7aHbaadaGadaqaaiaad6gaciGGSbGaaiOBamaadmaabaGaeu4K dC0aaeWaaeaadaWcaaqaaiaaigdaaeaacaWGRbaaaiabgUcaRiaaig daaiaawIcacaGLPaaadaaeWbqaamaalaaabaGaaiikaiabgkHiTiaa igdacaGGPaWaaWbaaeqajuaibaGaamOAaaaajuaGdaqadaqaamaaDa aabaGaamOAaaqaaiabeg7aHjabgkHiTiaaigdaaaaacaGLOaGaayzk aaaabaWaaeWaaeaacaaIXaGaey4kaSIaamOAaaGaayjkaiaawMcaam aaCaaabeqaamaalaaabaGaaGymaaqaaiaadUgaaaGaey4kaSIaaGym aaaaaaaajuaibaGaamOAaiabg2da9iaaicdaaeaacqGHEisPaKqbak abggHiLdaacaGLBbGaayzxaaaacaGL7bGaayzFaaGaeyypa0JaaGim aaaa@CCC0@


.… (23)
…. λ L(α,λ,k)= (k+1) λ + k x i k λ k+1 i=1 n { (α1)exp{(k x i k / λ k+1 )} 1exp{ ( x i /λ) k } } α1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITaeaacqGHciITcqaH7oaBaaGaamitaiaacIcacqaHXoqy caGGSaGaeq4UdWMaaiilaiaadUgacaGGPaGaeyypa0JaeyOeI0YaaS aaaeaacaGGOaGaam4AaiabgUcaRiaaigdacaGGPaaabaGaeq4UdWga aiabgUcaRmaalaaabaGaam4AaiaadIhadaWgaaqcfasaaiaadMgaae qaaKqbaoaaCaaajuaibeqaaiaadUgaaaaajuaGbaGaeq4UdW2aaWba aeqajuaibaGaam4AaiabgUcaRiaaigdaaaaaaKqbakabgkHiTmaaqa habaWaaiWaaeaadaWcaaqaaiaacIcacqaHXoqycqGHsislcaaIXaGa aiykaiGacwgacaGG4bGaaiiCaiaacUhacqGHsislcaGGOaGaam4Aai aadIhadaWgaaqcfasaaiaadMgaaeqaaKqbaoaaCaaajuaibeqaaiaa dUgaaaqcfaOaai4laiabeU7aSnaaCaaabeqcfasaaiaadUgacqGHRa WkcaaIXaaaaKqbakaacMcacaGG9baabaGaaGymaiabgkHiTiGacwga caGG4bGaaiiCaiaacUhacqGHsislcaGGOaGaamiEamaaBaaajuaiba GaamyAaaqabaqcfaOaai4laiabeU7aSjaacMcadaahaaqabKqbGeaa caWGRbaaaKqbakaac2haaaaacaGL7bGaayzFaaaajuaibaGaamyAai abg2da9iaaigdaaeaacaWGUbaajuaGcqGHris5amaaCaaabeqcfasa aiabeg7aHjabgkHiTiaaigdaaaqcfaOaeyypa0JaaGimaaaa@8C8F@  ………… (24)

Also, equations (22), (23) and (24) are solved iteratively to obtain the maximum likelihood estimates of the parameters

α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ , k and λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ .

AIC and log-likelihood

We calculate AIC value for each model with the same dataset, and the best model is the one with minimum AIC value. The value of AIC depends on the data Pines and Bombax, which leads to model selection uncertainty.
AIC=2logL( θ ^ | x i )+2k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqai aadMeacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGVbGaai4z aiaadYeacaGGOaWaaCbiaeaacuaH4oqCgaqcaaqabKqbGeaaaaqcfa OaaiiFaiaadIhadaWgaaqcfasaaiaadMgaaKqbagqaaiaacMcacqGH RaWkcaaIYaGaam4Aaaaa@499F@
where

  • L( θ Λ | x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai aacIcadaWfGaqaaiabeI7aXbqabKqbGeaacqqHBoataaqcfaOaaiiF aiaadIhadaWgaaqcfasaaiaadMgaaeqaaKqbakaacMcaaaa@409B@ = the maximized value of the likelihood function of the model, and where θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbiae aacuaH4oqCgaqcaaqabKqbGeaaaaaaaa@38B6@  are the parameter values that maximize the likelihood function;
  • x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamyAaaqcfayabaaaaa@394C@ = the observed data;
  • k = the number of free parameters to be estimated.

Results and discussion

Summary of the data    

The Bombax and Pine Height-Diameter data were extracted from the Forestry Research Institute of Nigeria’s records, cleaned up and the summary statistics of the data was computed as presented in Tables 1 and 2.

PinesHEIGHT

PinesDBH

Bombaxheight

Bombaxdbh

Mean

13.33333

13.87566

8.82142

16.47857

estimated Stdev

3.38336

3.84751

2.67723

6.37982

estimated skewness

-0.14219

0.11252

0.6269

1.22949

estimated kurtosis

2.683

2.76197

2.85557

5.31719

Table 1 Descriptive statistics of the data

EW

PinesHEIGHT    PinesDBH   Bombaxheight    Bombaxdbh

Mean
estimated stdev
estimated skewness
estimated kurtosis

0.02165                  0.046100        0.008641               0.008940 0.00685                0.013384              0.044389        0.066902
-0.77246              -1.379207       5.268808               7.281359
2.757014              3.945957        29.65974               54.01818

LBEW
Mean
estimated stdev
estimated skewness
estimated kurtosis

 

6.34E-05               7.07E-05        3.32E-05               5.88E-05              4.28E-05              4.78E-05               2.62E-05       4.82E-05
2.408499              2.212744       2.871012              3.017609
10.28893              9.160734       15.27437              15.87016

ABEW
Mean
estimated stdev
estimated skewness
estimated kurtosis

 

8.97E-05               4.76E-05        0.000130              4.88E-05 
5.58E-05               3.51E-05        7.95E-05               4.20E-05
2.299099               2.336502        2.215247             3.142198
9.618972               9.925224        11.11609             16.72621

Table 2 Skewness and kurtosis of EW, LBEW and ABEW distribution

Maximum likelihood approach

The above Table 3-5 shown the parameters estimation of EW, LBEW and ABEW distributions. We observed the comparison of the three distributions by their corresponding AIC and -2log-likelihood of each of the dataset pines and bombax. The ideal distribution is the one with the minimum AIC values (Figures 1-5).7-15

EW

Pines

Bombax

parameters

HEIGHT

DBH

Height

dbh

k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E6@
α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3795@
λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWgaaa@37AA@
AIC
-2loglik

4.351
1.053
14.468
1999.52
993.76

3.077
1.635
13.542
2098.4
1043.2

1.433
8.920
4.352
538.40
263.20

1.064
11.725
5.747
719.45
353.72

Table 3 Parameters estimation of EW

LBEW

Pines

Bombax

parameters

HEIGHT

DBH

Height

dbh

k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E6@

0.05

0.05

0.05

0.05

α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3795@

0.264

0.261

0.263

0.261

λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWgaaa@37AA@

0.045

0.052

0.01

0.053

AIC

1629.07

1634.36

467.19

484.93

-2loglik

407.117

408.44

116.65

121.08

Table 4 Parameters estimation of LBEW

ABEW

Pines

Bombax

parameters

HEIGHT

DBH

Height

dbh

k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E6@
α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3795@
λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWgaaa@37AA@
AIC
-2loglik

0.050
0.262
0.068
1533.11
383.128

0.050
0.262
0.034
1508.10
376.874

0.050
0.267
0.045
449.67
112.27  

0.050
0.263
0.045
450.24
112.41

Table 5 Parameters estimation of ABEW

Figure 1 Histogram boxplot plot of bombax and pines H-D.

Figure 2 The probability distribution function of the EW, LBEW and ABEW distribution.

Figure 3 The cumulative distribution function of the EW distribution.

Figure 4 The cumulative distribution function of the LBEW distribution.

Figure 5 The cumulative distribution function of the ABEW distribution.

Conclusion

This study introduced a new distribution based on LBEW and ABEW. Some characteristics of the new distributions were obtained. Plots for the cumulative distribution function, pdf and tables with values of skewness and kurtosis were also provided. Height-Diameter (H-D) data on Bombax and Pines (Pinus caribeae) were used to demonstrate the application of the distributions. Estimation of parameters of EW, LBEW and ABEW distributions were done using the maximum likelihood approach and compared across the distributions using criteria like AIC and Log-likelihood. We therefore proposed that similar to Exponentiated Weibull distribution (EW), a better fitting of Bombax and Pines H-D data are possible by LBEW and ABEW distributions.

Acknowledgements

We gratefully acknowledge the suggestions given by the anonymous referees, which have immensely helped to improve the presentation of the paper.

Conflicts of interest

None.

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