eISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 7 Issue 3

Area–biased poisson exponential distribution with applications

Ayesha Fazal
Roll of honor Mphil statistics, Kinnaird College for women, Pakistan
Received: May 24, 2018 | Published: June 28, 2018

Correspondence: Ayesha Fazal, Department of statistics, Roll of honor Mphil statistics, Kinnaird College for women Lahore, Pakistan, Email

Citation: Fazal A. Area–biased poisson exponential distribution with applications. Biom Biostat Int J. 2018;7(3):256‒261. DOI: 10.15406/bbij.2018.07.00216

Abstract

In this paper, an area–biased form of the single parameter Poisson exponential distribution (PED) is obtained by area biasing the discrete Poisson exponential distribution (PED) introduced by Fazal & Bashir.1 Poisson–exponential distribution is an important discrete distribution which has many applications in countable datasets. The first four moments (about origin) and the central moments (about mean) have been obtained and hence expression for coefficient of variation (CV), skewness, kurtosis and index of dispersion are derived. To estimate the parameters of Area–biased Poisson exponential distribution (ABPED), maximum likelihood method (MLE) and method of moments (MOM) are also developed. The goodness of fit for (ABPED) has been discussed using three real data–sets the fit shows a better fit over size–biased Poisson Lindley distribution (SBPLD).

Keywords: poisson exponential distribution, weighted distributions, moments, estimation of parameters, goodness of fit

Introduction

Fazal & Bashir1 have obtained discrete Poisson exponential distribution (PED) for modelling count data with probability mass function (p.m.f)

P(X=x)= θ (1+θ) x+1 ,θ>0,x=0,1,2, ...  ... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaamiuaiaacIcacaWGybGaeyypa0JaamiEaiaacMcacqGH 9aqpjuaGdaWcaaqaaKqzGeGaeqiUdehajuaGbaqcLbsacaGGOaGaaG ymaiabgUcaRiabeI7aXjaacMcajuaGdaahaaqabKqbGeaajugWaiaa dIhacqGHRaWkcaaIXaaaaaaajugibiaacYcacqaH4oqCcqGH+aGpca aIWaGaaiilaiaadIhacqGH9aqpcaaIWaGaaiilaiaaigdacaGGSaGa aGOmaiaacYcacaGGGcGaaiOlaiaac6cacaGGUaGaaiiOaiaacckaca GGUaGaaiOlaiaac6caaaa@5ECB@ (1.1)

The Poisson exponential distribution (PED) in (1.1) is a mixture of Poisson and exponential distribution when the parameter of Poisson distribution (λ) follows exponential distribution. The first four moments (about origin) and the variance of (PED) obtained by Fazal & Bashir1 are given as

μ 1 = 1 θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGafqiVd02dayaafaaddaWgaaqcbasaaKqzadWdbiaaigda aKqaG8aabeaajugib8qacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbi aaigdaaOWdaeaajugib8qacqaH4oqCaaaaaa@42A1@

  μ ' 2 = 2+θ θ 2   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaaiiOaiabeY7aTXWdamaaCaaajeaibeqaaKqzadWdbiaa cEcaaaadpaWaaSbaaKqaGeaajugWa8qacaaIYaaajeaipaqabaqcLb sapeGaeyypa0tcfa4aaSaaaOWdaeaajugib8qacaaIYaGaey4kaSIa eqiUdehak8aabaqcLbsapeGaeqiUdexcfa4damaaCaaaleqajeaiba qcLbmapeGaaGOmaaaaaaqcLbsacaGGGcaaaa@4D44@ (1.2)

μ 3 = θ 2 +6θ+6 θ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGafqiVd02dayaafaaddaWgaaqcbasaaKqzadWdbiaaioda aKqaG8aabeaajugib8qacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbi abeI7aXLqba+aadaahaaWcbeqcbasaaKqzadWdbiaaikdaaaqcLbsa cqGHRaWkcaaI2aGaeqiUdeNaey4kaSIaaGOnaaGcpaqaaKqzGeWdbi abeI7aXLqba+aadaahaaWcbeqcbasaaKqzadWdbiaaiodaaaaaaaaa @4F04@

μ 4 = θ 3 +14 θ 2 +36θ+24 θ 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGafqiVd02dayaafaaddaWgaaqcbasaaKqzadWdbiaaisda aKqaG8aabeaajugib8qacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbi abeI7aXLqba+aadaahaaWcbeqcbasaaKqzadWdbiaaiodaaaqcLbsa cqGHRaWkcaaIXaGaaGinaiabeI7aXLqba+aadaahaaWcbeqcbasaaK qzadWdbiaaikdaaaqcLbsacqGHRaWkcaaIZaGaaGOnaiabeI7aXjab gUcaRiaaikdacaaI0aaak8aabaqcLbsapeGaeqiUdexcfa4damaaCa aaleqajqwaa+FaaKqzGcWdbiaaisdaaaaaaaaa@59AF@

μ 2 = 1+θ θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaeqiVd0wcfa4damaaBaaajeaibaqcLbmapeGaaGOmaaWc paqabaqcLbsapeGaeyypa0tcfa4aaSaaaOWdaeaajugib8qacaaIXa Gaey4kaSIaeqiUdehak8aabaqcLbsapeGaeqiUdexcfa4damaaCaaa leqajeaibaqcLbmapeGaaGOmaaaaaaaaaa@487F@

The mathematical properties and estimation of parameter have been discussed by Fazal & Bashir1 and its application proves that it is a good replacement of poisson distribution and Lindley distribution. The size–biased form of (PED) has been discussed by Fazal & Bashir1 and its goodness of fit gives quite satisfactory fit over size–biased poisson distribution, size–biased lindley distribution and size–biased geometric distribution. The mixture of Poisson and Size–biased exponential distribution has been discussed by Fazal & Bashir1 with properties and applications.

The size–biased and Area–biased distributions were discussed earlier by Fisher2 when sample observations have unequal probability of selection, therefore we apply weights to the distribution to model bias.

If the rv ‘x’ had pdf f(x,θ); x=0,1,2,…..,; θ>0 , then the weighted distribution is of the form

P( x;θ )= x m f o ( x;θ ) μ m        MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaamiuaKqbaoaabmaak8aabaqcLbsapeGaamiEaiaacUda cqaH4oqCaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaSaaaOWdae aajugib8qacaWG4bqcfa4damaaCaaaleqajqwaa+FaaKqzGcWdbiaa d2gaaaqcLbsacaWGMbqcfa4damaaBaaajeaibaqcLbmapeGaam4Baa Wcpaqabaqcfa4dbmaabmaak8aabaqcLbsapeGaamiEaiaacUdacqaH 4oqCaOGaayjkaiaawMcaaaWdaeaajugib8qacuaH8oqBpaGbauaamm aaBaaajeaibaqcLbmapeGaamyBaaqcbaYdaeqaaaaajugib8qacaGG GcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaaaa@5FB1@ (1.3)

For m=1 and m=2 we get the size–biased and area–biased distributions respectively. Area–biased distributions are applicable for sampling in forestry, medical sciences, psychology etc, different discrete mixed distributions have been size–biased and discussed with their applications in real data–sets. Shankar & Kumar3 obtained size–biased Poisson Garima distribution with mathematical properties to analyze genetics datasets. Shankar4 introduced size–biased Poisson Shankar distribution with applications. Shankar & Fasshaye5 considered the size–biased form of Poisson Sujata distribution which was first introduced by Shankar5 for modelling count data in various fields of knowledge. Shakila & Mujahid Rasul6 derived the Poisson Area–Biased Lindley distribution with its applications in biological data to prove that it gives a better fit than Poisson distribution. Shankar & Fassahe7 proposed the size–biased form of Poisson Amarenda distribution and its applications proved that it is a good replacement of size–biased Poisson distribution (SBPD) size–biased Poisson Lindley distribution (SBPLD) and Size–biased Poisson.

Sujhata Distribution (SBPSD).Shakila and Mujahid8 proposed the size–biased form of Poisson Janardhan distribution and derived its mathematical properties, whereas Janardhan distribution is a two parameter distribution obtained by Rama & Mishra9 as a mixture of exponential and gamma distribution. Rama & Mishra9 obtained the size–biased form of Qaussi Poisson–Lindley distribution of which size–biased poisson Lindley distribution is a particular case (SBPLD). Ahmed & Munir10 have discussed few size–biased discrete distributions and their generalizations with properties and application. The size–biased version of Poisson Lindley distribution has been discussed by Ghitanni & Mutairi11 and the new distribution introduced in this paper i.e Area–biased poisson exponential distribution (ABPED) gives more satisfactory fit as compared to size–biased Poisson Lindley distribution. The mathematical properties and estimation of parameters has been discussed and goodness of fit is also presented.12–16

Area–biased poisson exponential distribution

Using (1.1), (1.2), (1.3) the pmf of the Area–biased Poisson exponential distribution can be obtained as

P(x;θ)= x 2 f o ( x;θ ) μ 2 = x 2 θ/ ( 1+θ ) x+1 ( 2+θ )/ θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaamiuaiaacIcacaWG4bGaai4oaiabeI7aXjaacMcacqGH 9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiaadIhajuaGpaWaaWbaaSqabK qaGeaajugWa8qacaaIYaaaaKqzGeGaamOzaKqba+aadaWgaaWcbaqc LbsapeGaam4BaaWcpaqabaqcfa4dbmaabmaak8aabaqcLbsapeGaam iEaiaacUdacqaH4oqCaOGaayjkaiaawMcaaaWdaeaajugib8qacuaH 8oqBpaGbauaammaaBaaajeaibaqcLbmapeGaaGOmaaqcbaYdaeqaaa aajugib8qacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiaadIhajuaG paWaaWbaaSqabKqaGeaajugWa8qacaaIYaaaaKqzGeGaeqiUdeNaai 4laKqbaoaabmaak8aabaqcLbsapeGaaGymaiabgUcaRiabeI7aXbGc caGLOaGaayzkaaqcfa4damaaCaaaleqajeaibaqcLbmapeGaamiEai abgUcaRiaaigdaaaaak8aabaqcfa4dbmaabmaak8aabaqcLbsapeGa aGOmaiabgUcaRiabeI7aXbGccaGLOaGaayzkaaqcLbsacaGGVaGaeq iUdexcfa4damaaCaaaleqajeaibaqcLbmapeGaaGOmaaaaaaaaaa@746F@

Where μ 2 = ( 2+θ ) θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGafqiVd02dayaafaaddaWgaaqcbasaaKqzadWdbiaaikda aKqaG8aabeaajugib8qacqGH9aqpjuaGdaWcaaGcpaqaaKqba+qada qadaGcpaqaaKqzGeWdbiaaikdacqGHRaWkcqaH4oqCaOGaayjkaiaa wMcaaaWdaeaajugib8qacqaH4oqCjuaGpaWaaWbaaSqabKqaGeaaju gWa8qacaaIYaaaaaaaaaa@4A69@  is the second raw moment of discrete poisson exponential distribution.

With simplifications we get the pmf of Area–biased Poisson exponential distribution with parameter θ as

P( X=x )= x 2 θ 3 ( 1+θ ) x+1 ( 2+θ )                θ>0  ,   x=1,2,3,4,.    MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaamiuaKqbaoaabmaak8aabaqcLbsapeGaamiwaiabg2da 9iaadIhaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaSaaaOWdae aajugib8qacaWG4bqcfa4damaaCaaaleqajeaibaqcLbmapeGaaGOm aaaajugibiabeI7aXLqba+aadaahaaWcbeqcbasaaKqzadWdbiaaio daaaaak8aabaqcfa4dbmaabmaak8aabaqcLbsapeGaaGymaiabgUca RiabeI7aXbGccaGLOaGaayzkaaqcfa4damaaCaaaleqajeaibaqcLb mapeGaamiEaiabgUcaRiaaigdaaaqcfa4aaeWaaOWdaeaajugib8qa caaIYaGaey4kaSIaeqiUdehakiaawIcacaGLPaaaaaqcLbsacaGGGc GaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaaccka caGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacqaH4oqCcqGH+a GpcaaIWaGaaiiOaiaacckacaGGSaGaaiiOaiaacckacaGGGcGaamiE aiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaaiilaiaais dacaGGSaGaeyOjGWRaeyOjGWRaeyOjGWRaaiOlaiaacckacaGGGcGa aiiOaaaa@880D@ (1.4)

                 

Graphs of Area– biased Poisson exponential distribution for different values of θ are shown in Figure 1 below

Figure 1 Graphs of Area–biased Poisson exponential distribution for different values of θ.

Moments and moment based measures of area–biased poisson exponential distribution

We start the mathematical derivations with moments and moment measures.
The first four raw moments of Area–Biased Poisson exponential distribution (ABPED) are

μ 1 = θ 2 +6θ+θ6 θ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGafqiVd02dayaafaaddaWgaaqcbasaaKqzadWdbiaaigda aKqaG8aabeaajugib8qacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbi abeI7aXLqba+aadaahaaWcbeqcbasaaKqzadWdbiaaikdaaaqcLbsa cqGHRaWkcaaI2aGaeqiUdeNaey4kaSIaeqiUdeNaaGOnaaGcpaqaaK qzGeWdbiabeI7aXLqbaoaabmaak8aabaqcLbsapeGaeqiUdeNaey4k aSIaaGOmaaGccaGLOaGaayzkaaaaaaaa@53F6@

μ 2 = θ 2 +12θ+12 θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGafqiVd02dayaafaaddaWgaaqcbasaaKqzadWdbiaaikda aKqaG8aabeaajugib8qacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbi abeI7aXLqba+aadaahaaWcbeqcbasaaKqzadWdbiaaikdaaaqcLbsa cqGHRaWkcaaIXaGaaGOmaiabeI7aXjabgUcaRiaaigdacaaIYaaak8 aabaqcLbsapeGaeqiUdexcfa4damaaCaaaleqajeaibaqcLbmapeGa aGOmaaaaaaaaaa@5070@

μ 3 = θ 4 +30 θ 3 +150 θ 2 +240θ+120 θ 3 ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGafqiVd02dayaafaaddaWgaaqcbasaaKqzadWdbiaaioda aKqaG8aabeaajugib8qacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbi abeI7aXLqba+aadaahaaWcbeqcbasaaKqzadWdbiaaisdaaaqcLbsa cqGHRaWkcaaIZaGaaGimaiabeI7aXLqba+aadaahaaWcbeqcbasaaK qzadWdbiaaiodaaaqcLbsacqGHRaWkcaaIXaGaaGynaiaaicdacqaH 4oqCjuaGpaWaaWbaaSqabKqaGeaajugWa8qacaaIYaaaaKqzGeGaey 4kaSIaaGOmaiaaisdacaaIWaGaeqiUdeNaey4kaSIaaGymaiaaikda caaIWaaak8aabaqcLbsapeGaeqiUdexcfa4damaaCaaaleqajeaiba qcLbmapeGaaG4maaaajuaGdaqadaGcpaqaaKqzGeWdbiabeI7aXjab gUcaRiaaikdaaOGaayjkaiaawMcaaaaaaaa@67EE@

μ 4 = θ 4 +60 θ 3 +420 θ 2 +720θ+360 θ 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGafqiVd02dayaafaaddaWgaaqcbasaaKqzadWdbiaaisda aKqaG8aabeaajugib8qacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbi abeI7aXLqba+aadaahaaWcbeqcbasaaKqzadWdbiaaisdaaaqcLbsa cqGHRaWkcaaI2aGaaGimaiabeI7aXLqba+aadaahaaWcbeqcbasaaK qzadWdbiaaiodaaaqcLbsacqGHRaWkcaaI0aGaaGOmaiaaicdacqaH 4oqCjuaGpaWaaWbaaSqabKqaGeaajugWa8qacaaIYaaaaKqzGeGaey 4kaSIaaG4naiaaikdacaaIWaGaeqiUdeNaey4kaSIaaG4maiaaiAda caaIWaaak8aabaqcLbsapeGaeqiUdexcfa4damaaCaaaleqajeaiba qcLbmapeGaaGinaaaaaaaaaa@61CF@

The mean moments of ABPED are obtained by using the relationship between moments about mean and moments about origin

μ 2 = 4 θ 3 +16 θ 2 24θ+12 θ 2 ( θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaeqiVd0wcfa4damaaBaaajeaibaqcLbmapeGaaGOmaaWc paqabaqcLbsapeGaeyypa0tcfa4aaSaaaOWdaeaajugib8qacaaI0a GaeqiUdexcfa4damaaCaaaleqajeaibaqcLbmapeGaaG4maaaajugi biabgUcaRiaaigdacaaI2aGaeqiUdexcfa4damaaCaaaleqajeaiba qcLbmapeGaaGOmaaaajugibiabgkHiTiaaikdacaaI0aGaeqiUdeNa ey4kaSIaaGymaiaaikdaaOWdaeaajugib8qacqaH4oqCjuaGpaWaaW baaSqabKqaafaajug4a8qacaaIYaaaaKqbaoaabmaak8aabaqcLbsa peGaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaqcfa4damaaCa aaleqajeaibaqcLbmapeGaaGOmaaaaaaaaaa@627F@

μ 3 = 4 θ 5 +28 θ 4 336 θ 3 +168 θ 2 +144θ+48 θ 3 ( θ+2 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaeqiVd0wcfa4damaaBaaajeaibaqcLbmapeGaaG4maaWc paqabaqcLbsapeGaeyypa0tcfa4aaSaaaOWdaeaajugib8qacaaI0a GaeqiUdexcfa4damaaCaaaleqajeaibaqcLbmapeGaaGynaaaajugi biabgUcaRiaaikdacaaI4aGaeqiUdexcfa4damaaCaaaleqajeaiba qcLbmapeGaaGinaaaajugibiabgkHiTiaaiodacaaIZaGaaGOnaiab eI7aXLqba+aadaahaaWcbeqcbasaaKqzadWdbiaaiodaaaqcLbsacq GHRaWkcaaIXaGaaGOnaiaaiIdacqaH4oqCjuaGpaWaaWbaaSqabKqa GeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaaGymaiaaisdacaaI0a GaeqiUdeNaey4kaSIaaGinaiaaiIdaaOWdaeaajugib8qacqaH4oqC juaGpaWaaWbaaSqabKqaGeaajugWa8qacaaIZaaaaKqbaoaabmaak8 aabaqcLbsapeGaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaqc fa4damaaCaaaleqajeaqbaqcLboapeGaaG4maaaaaaaaaa@73F1@

μ 4 = 760 θ 7 +938 θ 6 +6200 θ 5 +32992 θ 4 +97632 θ 3 +132672 θ 2 +65088θ+720 θ 4 ( θ+2 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaeqiVd0wcfa4damaaBaaajqwaa+FaaKqzGcWdbiaaisda aSWdaeqaaKqzGeWdbiabg2da9Kqbaoaalaaak8aabaqcLbsapeGaaG 4naiaaiAdacaaIWaGaeqiUdexcfa4damaaCaaaleqajeaibaqcLbma peGaaG4naaaajugibiabgUcaRiaaiMdacaaIZaGaaGioaiabeI7aXL qba+aadaahaaWcbeqcbasaaKqzadWdbiaaiAdaaaqcLbsacqGHRaWk caaI2aGaaGOmaiaaicdacaaIWaGaeqiUdexcfa4damaaCaaaleqaje aibaqcLbmapeGaaGynaaaajugibiabgUcaRiaaiodacaaIYaGaaGyo aiaaiMdacaaIYaGaeqiUdexcfa4damaaCaaaleqajeaibaqcLbmape GaaGinaaaajugibiabgUcaRiaaiMdacaaI3aGaaGOnaiaaiodacaaI YaGaeqiUdexcfa4damaaCaaaleqajeaibaqcLbmapeGaaG4maaaaju gibiabgUcaRiaaigdacaaIZaGaaGOmaiaaiAdacaaI3aGaaGOmaiab eI7aXLqba+aadaahaaWcbeqcbasaaKqzadWdbiaaikdaaaqcLbsacq GHRaWkcaaI2aGaaGynaiaaicdacaaI4aGaaGioaiabeI7aXjabgUca RiaaiEdacaaIYaGaaGimaaGcpaqaaKqzGeWdbiabeI7aXLqba+aada ahaaWcbeqcbasaaKqzadWdbiaaisdaaaqcfa4aaeWaaOWdaeaajugi b8qacqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaajuaGpaWaaW baaSqabKqaGeaajugWa8qacaaI0aaaaaaaaaa@905F@

The Harmonic mean of Area–biased Poisson exponential distribution is

H.M= θ θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaamisaiaac6cacaWGnbGaeyypa0tcfa4aaSaaaOWdaeaa jugib8qacqaH4oqCaOWdaeaajugib8qacqaH4oqCcqGHRaWkcaaIYa aaaaaa@4292@

The coefficient of variation (C.V), coefficient of skewness ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqaaaaa aaaaWdbmaabmaak8aabaqcfa4dbmaakaaak8aabaqcLbsapeGaeqOS diwcfa4damaaBaaajeaibaqcLbmapeGaaGymaaWcpaqabaaapeqaba aakiaawIcacaGLPaaaaaa@3FE0@ , coefficient of kurtosis ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqaaaaa aaaaWdbmaabmaak8aabaqcfa4dbmaakaaak8aabaqcLbsapeGaeqOS diwcfa4damaaBaaajeaibaqcLbmapeGaaGymaaWcpaqabaaapeqaba aakiaawIcacaGLPaaaaaa@3FE0@  and index of dispersion (γ) of Area–biased Poisson exponential distribution ABPED are obtained as:

C.V= σ μ 1 = 4 θ 3 +16 θ 2 24θ+12 θ 2 +6θ+6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaam4qaiaac6cacaWGwbGaeyypa0tcfa4aaSaaaOWdaeaa jugib8qacqaHdpWCaOWdaeaajugib8qacuaH8oqBpaGbauaammaaBa aajeaibaqcLbmapeGaaGymaaqcbaYdaeqaaaaajugib8qacqGH9aqp juaGdaWcaaGcpaqaaKqba+qadaGcaaGcpaqaaKqzGeWdbiaaisdacq aH4oqCjuaGpaWaaWbaaSqabKqaGeaajugWa8qacaaIZaaaaKqzGeGa ey4kaSIaaGymaiaaiAdacqaH4oqCjuaGpaWaaWbaaSqabKqaGeaaju gWa8qacaaIYaaaaKqzGeGaeyOeI0IaaGOmaiaaisdacqaH4oqCcqGH RaWkcaaIXaGaaGOmaaWcbeaaaOWdaeaajugib8qacqaH4oqCjuaGpa WaaWbaaSqabKqaGeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaaGOn aiabeI7aXjabgUcaRiaaiAdaaaaaaa@6653@

β 1 = μ 3 μ 2 3 2 = 4 θ 5 +28 θ 4 336 θ 3 +168 θ 2 +144θ+48 ( 4 θ 3 +16 θ 2 24θ+12 ) 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqaaaaa aaaaWdbmaakaaak8aabaqcLbsapeGaeqOSdiwcfa4damaaBaaajeai baqcLbmapeGaaGymaaWcpaqabaaapeqabaqcLbsacqGH9aqpjuaGda WcaaGcpaqaaKqzGeWdbiabeY7aTLqba+aadaWgaaqcbasaaKqzadWd biaaiodaaSWdaeqaaaGcbaqcLbsapeGaeqiVd0wcfa4damaaBaaaje aibaqcLbmapeGaaGOmaaWcpaqabaqcfa4aaWbaaSqabKqaGeaam8qa daWccaqcbaYdaeaajugWa8qacaaIZaaajeaipaqaaKqzadWdbiaaik daaaaaaaaajugibiabg2da9Kqbaoaalaaak8aabaqcfa4dbmaakaaa k8aabaqcLbsapeGaaGinaiabeI7aXLqba+aadaahaaWcbeqcbasaaK qzadWdbiaaiwdaaaqcLbsacqGHRaWkcaaIYaGaaGioaiabeI7aXLqb a+aadaahaaWcbeqcbasaaKqzadWdbiaaisdaaaqcLbsacqGHsislca aIZaGaaG4maiaaiAdacqaH4oqCjuaGpaWaaWbaaSqabKqaGeaajugW a8qacaaIZaaaaKqzGeGaey4kaSIaaGymaiaaiAdacaaI4aGaeqiUde xcfa4damaaCaaaleqajeaibaqcLbmapeGaaGOmaaaajugibiabgUca RiaaigdacaaI0aGaaGinaiabeI7aXjabgUcaRiaaisdacaaI4aaale qaaaGcpaqaaKqba+qadaqadaGcpaqaaKqzGeWdbiaaisdacqaH4oqC juaGpaWaaWbaaSqabKqaGeaajugWa8qacaaIZaaaaKqzGeGaey4kaS IaaGymaiaaiAdacqaH4oqCjuaGpaWaaWbaaSqabKqaGeaajugWa8qa caaIYaaaaKqzGeGaeyOeI0IaaGOmaiaaisdacqaH4oqCcqGHRaWkca aIXaGaaGOmaaGccaGLOaGaayzkaaqcfa4damaaCaaaleqajeaibaad peWaaSGaaKqaG8aabaqcLbmapeGaaG4maaqcbaYdaeaajugWa8qaca aIYaaaaaaaaaaaaa@95B0@

β 2 = μ 4 μ 2 2 = 76 θ 7 +938 θ 6 +6200 θ 5 +32992 θ 4 +97632 θ 3 +132672 θ 2 +65088θ+720 ( 4 θ 3 +16 θ 2 24θ+12 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaeqOSdiwcfa4damaaBaaajeaibaqcLbmapeGaaGOmaaWc paqabaqcLbsapeGaeyypa0tcfa4aaSaaaOWdaeaajugib8qacqaH8o qBjuaGpaWaaSbaaKqaGeaajugWa8qacaaI0aaal8aabeaaaOqaaKqz GeWdbiabeY7aTLqba+aadaWgaaqcbasaaKqzadWdbiaaikdaaSWdae qaaKqbaoaaCaaaleqajeaibaqcLbmapeGaaGOmaaaaaaqcLbsacqGH 9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiaaiEdacaaI2aGaeqiUdexcfa 4damaaCaaaleqajeaibaqcLbmapeGaaG4naaaajugibiabgUcaRiaa iMdacaaIZaGaaGioaiabeI7aXLqba+aadaahaaWcbeqcbasaaKqzad WdbiaaiAdaaaqcLbsacqGHRaWkcaaI2aGaaGOmaiaaicdacaaIWaGa eqiUdexcfa4damaaCaaaleqajeaibaqcLbmapeGaaGynaaaajugibi abgUcaRiaaiodacaaIYaGaaGyoaiaaiMdacaaIYaGaeqiUdexcfa4d amaaCaaaleqajeaibaqcLbmapeGaaGinaaaajugibiabgUcaRiaaiM dacaaI3aGaaGOnaiaaiodacaaIYaGaeqiUdexcfa4damaaCaaaleqa jeaibaqcLbmapeGaaG4maaaajugibiabgUcaRiaaigdacaaIZaGaaG OmaiaaiAdacaaI3aGaaGOmaiabeI7aXLqba+aadaahaaWcbeqcbasa aKqzadWdbiaaikdaaaqcLbsacqGHRaWkcaaI2aGaaGynaiaaicdaca aI4aGaaGioaiabeI7aXjabgUcaRiaaiEdacaaIYaGaaGimaaGcpaqa aKqba+qadaqadaGcpaqaaKqzGeWdbiaaisdacqaH4oqCjuaGpaWaaW baaSqabKqaGeaajugWa8qacaaIZaaaaKqzGeGaey4kaSIaaGymaiaa iAdacqaH4oqCjuaGpaWaaWbaaSqabKqaGeaajugWa8qacaaIYaaaaK qzGeGaeyOeI0IaaGOmaiaaisdacqaH4oqCcqGHRaWkcaaIXaGaaGOm aaGccaGLOaGaayzkaaqcfa4damaaCaaaleqajeaibaqcLbmapeGaaG Omaaaaaaaaaa@A931@

γ= σ 2 μ 1 ' = 4 θ 3 +16 θ 2 24θ+12 θ( θ+2 )( θ 2 +6θ+6 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaeq4SdCMaeyypa0tcfa4aaSaaaOWdaeaajugib8qacqaH dpWCjuaGpaWaaWbaaSqabeaajugib8qacaaIYaaaaaGcpaqaaKqzGe WdbiabeY7aTXWdamaaBaaajeaibaqcLbmapeGaaGymaaqcbaYdaeqa aWWaaWbaaKqaGeqabaqcLbmapeGaai4jaaaaaaqcLbsacqGH9aqpju aGdaWcaaGcpaqaaKqzGeWdbiaaisdacqaH4oqCjuaGpaWaaWbaaSqa bKqaGeaajugWa8qacaaIZaaaaKqzGeGaey4kaSIaaGymaiaaiAdacq aH4oqCjuaGpaWaaWbaaSqabKqaGeaajugWa8qacaaIYaaaaKqzGeGa eyOeI0IaaGOmaiaaisdacqaH4oqCcqGHRaWkcaaIXaGaaGOmaaGcpa qaaKqzGeWdbiabeI7aXLqbaoaabmaak8aabaqcLbsapeGaeqiUdeNa ey4kaSIaaGOmaaGccaGLOaGaayzkaaqcfa4aaeWaaOWdaeaajugib8 qacqaH4oqCjuaGpaWaaWbaaSqabKqaGeaajugWa8qacaaIYaaaaKqz GeGaey4kaSIaaGOnaiabeI7aXjabgUcaRiaaiAdaaOGaayjkaiaawM caaaaaaaa@73D9@

For the Area–biased Poisson exponential distribution, (ABPED), from equations (1.6) and (1.7) it can be seen that ( γ 1 , β 2 )( 1.15, 5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaacI caqaaaaaaaaaWdbiabeo7aNLqbaoaaBaaajeaibaqcLbmacaaIXaaa leqaaKqzGeGaaiilaiabek7aILqba+aadaWgaaqcbasaaKqzadWdbi aaikdaaSWdaeqaaKqzGeGaaiyka8qacqGHsgIRjuaGpaWaaeWaaOqa aKqzGeWdbiaaigdacaGGUaGaaGymaiaaiwdacaGGSaGaaeiiaiaaiw daaOWdaiaawIcacaGLPaaaaaa@4E38@ as θ→0, the model is positively skewed and leptokurtic.

To study the characteristics and comparative behavior of ABPED and SBPLD, a table of ( γ 1 , β 2 )( 1.15, 5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaacI caqaaaaaaaaaWdbiabeo7aNLqbaoaaBaaajeaibaqcLbmacaaIXaaa leqaaKqzGeGaaiilaiabek7aILqba+aadaWgaaqcbasaaKqzadWdbi aaikdaaSWdaeqaaKqzGeGaaiyka8qacqGHsgIRjuaGpaWaaeWaaOqa aKqzGeWdbiaaigdacaGGUaGaaGymaiaaiwdacaGGSaGaaeiiaiaaiw daaOWdaiaawIcacaGLPaaaaaa@4E38@ for varying values of the parameter θ, has been prepared and presented in the table below (Table 1 & 2):

1

2

3

4

5

6

μ 1 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWajugiba baaaaaaaaapeGaa8hVdWWdamaaBaaajeaibaqcLbmapeGaaGymaaqc baYdaeqaaWWaaWbaaKqaGeqabaqcLbmapeGaa83jaaaaaaa@3E44@

4.3333

2.75

2.2

1.9167

1.7429

1.625

μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWajugiba baaaaaaaaapeGaa8hVdKqba+aadaWgaaqcbasaaKqzadWdbiaaikda aSWdaeqaaaaa@3C6A@

0.8889

0.5625

0.4267

0.3264

0.2547

0.2031

0.2176

0.3521

0.4199

0.4497

0.4614

0.4637

2.4749

2.3754

1.4434

0.7824

0.3155

0.03754

5254.97

621.85

245.63

155.66

121.86

106.53

0.2051

0.3409

0.3879

0.3877

0.37096

0.3494

Table 1 Values of θ for ABPED

 

1

2

3

4

5

6

μ 1 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWajugiba baaaaaaaaapeGaa8hVdWWdamaaBaaajeaibaqcLbmapeGaaGymaaqc baYdaeqaaWWaaWbaaKqaGeqabaqcLbmapeGaa83jaaaaaaa@3E44@

3.66667

2.25

1.8

1.583333

1.457143

1.375

μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWajugiba baaaaaaaaapeGaa8hVdKqba+aadaWgaaqcbasaaKqzadWdbiaaikda aSWdaeqaaaaa@3C6A@

5.55556

1.9375

1.093333

0.743056

0.556735

0.442708

0.642824

0.61864

0.580903

0.544425

0.512061

0.483901

1.318047

1.49478

1.649924

1.790721

1.921224

2.043701

5.4744

6.057232

6.599941

7.118613

7.625214

8.125813

1.515152

0.861111

0.607407

0.469298

0.382073

0.32197

Table 2 Values of θ for SBPLD

The comparative graphs of coefficient of variation, coefficient of skewness, coefficient of kurtosis and index of dispersion of ABPED and SBPLD are shown in Figure 2.

Figure 2 Graphs of for ABPED and SBPLD.

Reliability measures

Using pmf of ABPED from (1.4), we have

P( x+1,θ ) P( x.θ ) = ( 1+ 1 x ) 2 ( 1+θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqaaaaa aaaaWdbmaalaaak8aabaqcLbsapeGaamiuaKqbaoaabmaak8aabaqc LbsapeGaamiEaiabgUcaRiaaigdacaGGSaGaeqiUdehakiaawIcaca GLPaaaa8aabaqcLbsapeGaamiuaKqbaoaabmaak8aabaqcLbsapeGa amiEaiaac6cacqaH4oqCaOGaayjkaiaawMcaaaaajugibiabg2da9K qbaoaalaaak8aabaqcfa4dbmaabmaak8aabaqcLbsapeGaaGymaiab gUcaRKqbaoaaliaak8aabaqcLbsapeGaaGymaaGcpaqaaKqzGeWdbi aadIhaaaaakiaawIcacaGLPaaajuaGpaWaaWbaaSqabKqaGeaajugW a8qacaaIYaaaaaGcpaqaaKqba+qadaqadaGcpaqaaKqzGeWdbiaaig dacqGHRaWkcqaH4oqCaOGaayjkaiaawMcaaaaaaaa@5D36@

Which is a decreasing function of x, therefore ABPED is unimodal and has an increasing failure rate.

Generating functions of ABPED

Probability generating function: the probability generating function of ABPED can be obtained as

P x ( t )=E( t x )= x=1 x 2 θ 3 t x ( 1+θ ) x+1 ( 2+θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaamiuaKqba+aadaWgaaqcKfay=haajugWa8qacaWG4baa l8aabeaajuaGpeWaaeWaaOWdaeaajugib8qacaWG0baakiaawIcaca GLPaaajugibiabg2da9iaadweajuaGdaqadaGcpaqaaKqzGeWdbiaa dshajuaGpaWaaWbaaSqabKazba2=baqcLbmapeGaamiEaaaaaOGaay jkaiaawMcaaKqzGeGaeyypa0tcfa4aaabCaeaadaWcaaWdaeaajugi b8qacaWG4bqcfa4damaaCaaabeqcfasaaKqzadWdbiaaikdaaaqcLb sacqaH4oqCjuaGpaWaaWbaaeqajuaibaqcLbmapeGaaG4maaaajugi biaadshajuaGpaWaaWbaaeqajuaibaqcLbmapeGaamiEaaaaaKqba+ aabaWdbmaabmaapaqaaKqzGeWdbiaaigdacqGHRaWkcqaH4oqCaKqb akaawIcacaGLPaaapaWaaWbaaeqajuaibaqcLbmapeGaamiEaiabgU caRiaaigdaaaqcfa4aaeWaa8aabaqcLbsapeGaaGOmaiabgUcaRiab eI7aXbqcfaOaayjkaiaawMcaaaaaaKqbGeaajugWaiaadIhacqGH9a qpcaaIXaaajuaibaqcLbmacqGHEisPaKqzGeGaeyyeIuoaaaa@78A2@

= θ 3 t 2 +t( θ 4 + θ 3 ) θ 4 +5 θ 3 +9 θ 2 +7θ t 3 ( θ+2 )+3 t 2 ( θ 2 +3θ+2 )3t( θ 3 +4 θ 2 +5θ+2 )+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaeyypa0tcfa4aaSaaaOWdaeaajugib8qacqaH4oqCjuaG paWaaWbaaSqabKqaGeaajugWa8qacaaIZaaaaKqzGeGaamiDaKqba+ aadaahaaWcbeqcbasaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkcaWG 0bqcfa4aaeWaaOWdaeaajugib8qacqaH4oqCjuaGpaWaaWbaaSqabK qaGeaajugWa8qacaaI0aaaaKqzGeGaey4kaSIaeqiUdexcfa4damaa CaaaleqajeaibaqcLbmapeGaaG4maaaaaOGaayjkaiaawMcaaaWdae aajugib8qacqaH4oqCjuaGpaWaaWbaaSqabKqaGeaajugWa8qacaaI 0aaaaKqzGeGaey4kaSIaaGynaiabeI7aXLqba+aadaahaaWcbeqcba saaKqzadWdbiaaiodaaaqcLbsacqGHRaWkcaaI5aGaeqiUdexcfa4d amaaCaaaleqajeaibaqcLbmapeGaaGOmaaaajugibiabgUcaRiaaiE dacqaH4oqCcqGHsislcaWG0bqcfa4damaaCaaaleqajeaibaqcLbma peGaaG4maaaajuaGdaqadaGcpaqaaKqzGeWdbiabeI7aXjabgUcaRi aaikdaaOGaayjkaiaawMcaaKqzGeGaey4kaSIaaG4maiaadshajuaG paWaaWbaaSqabKqaGeaajugWa8qacaaIYaaaaKqbaoaabmaak8aaba qcLbsapeGaeqiUdexcfa4damaaCaaaleqajeaibaqcLbmapeGaaGOm aaaajugibiabgUcaRiaaiodacqaH4oqCcqGHRaWkcaaIYaaakiaawI cacaGLPaaajugibiabgkHiTiaaiodacaWG0bqcfa4aaeWaaOWdaeaa jugib8qacqaH4oqCjuaGpaWaaWbaaSqabKqaGeaajugWa8qacaaIZa aaaKqzGeGaey4kaSIaaGinaiabeI7aXLqba+aadaahaaWcbeqcbasa aKqzadWdbiaaikdaaaqcLbsacqGHRaWkcaaI1aGaeqiUdeNaey4kaS IaaGOmaaGccaGLOaGaayzkaaqcLbsacqGHRaWkcaaIYaaaaaaa@A1CB@

Moment generating function: the moment generating function of ABPED is obtained as

M x ( t )=E( e tx )= θ 3 e 2t +( θ 4 + θ 3 ) e t θ 4 +5 θ 3 +9 θ 2 +7θ( θ+2 ) e 3t +( 3 θ 2 +9θ+6 ) e 2t ( 3 θ 3 +12 θ 2 +15θ+6 ) e t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaamytaKqba+aadaWgaaqcbasaaKqzadWdbiaadIhaaSWd aeqaaKqba+qadaqadaGcpaqaaKqzGeWdbiaadshaaOGaayjkaiaawM caaKqzGeGaeyypa0JaamyraKqbaoaabmaak8aabaqcLbsapeGaamyz aKqba+aadaahaaWcbeqcbasaaKqzadWdbiaadshacaWG4baaaaGcca GLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiab eI7aXLqba+aadaahaaWcbeqcbasaaKqzadWdbiaaiodaaaqcLbsaca WGLbqcfa4damaaCaaaleqajeaibaqcLbmapeGaaGOmaiaadshaaaqc LbsacqGHRaWkjuaGdaqadaGcpaqaaKqzGeWdbiabeI7aXLqba+aada ahaaWcbeqcbasaaKqzadWdbiaaisdaaaqcLbsacqGHRaWkcqaH4oqC juaGpaWaaWbaaSqabKqaGeaajugWa8qacaaIZaaaaaGccaGLOaGaay zkaaqcLbsacaWGLbqcfa4damaaCaaaleqajeaibaqcLbmapeGaamiD aaaaaOWdaeaajugib8qacqaH4oqCjuaGpaWaaWbaaSqabKqaGeaaju gWa8qacaaI0aaaaKqzGeGaey4kaSIaaGynaiabeI7aXLqba+aadaah aaWcbeqcbasaaKqzadWdbiaaiodaaaqcLbsacqGHRaWkcaaI5aGaeq iUdexcfa4damaaCaaaleqajeaibaqcLbmapeGaaGOmaaaajugibiab gUcaRiaaiEdacqaH4oqCcqGHsisljuaGdaqadaGcpaqaaKqzGeWdbi abeI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaKqzGeGaamyzaKqb a+aadaahaaWcbeqcbasaaKqzadWdbiaaiodacaWG0baaaKqzGeGaey 4kaSscfa4aaeWaaOWdaeaajugib8qacaaIZaGaeqiUdexcfa4damaa CaaaleqajeaibaqcLbmapeGaaGOmaaaajugibiabgUcaRiaaiMdacq aH4oqCcqGHRaWkcaaI2aaakiaawIcacaGLPaaajugibiaadwgajuaG paWaaWbaaSqabKqaGeaajugWa8qacaaIYaGaamiDaaaajugibiabgk HiTKqbaoaabmaak8aabaqcLbsapeGaaG4maiabeI7aXLqba+aadaah aaWcbeqcbasaaKqzadWdbiaaiodaaaqcLbsacqGHRaWkcaaIXaGaaG OmaiabeI7aXLqba+aadaahaaWcbeqcbasaaKqzadWdbiaaikdaaaqc LbsacqGHRaWkcaaIXaGaaGynaiabeI7aXjabgUcaRiaaiAdaaOGaay jkaiaawMcaaKqzGeGaamyzaKqba+aadaahaaWcbeqcbauaaKqzGdWd biaadshaaaaaaaaa@BF67@

Fisher information matrix
If  x~f(x/θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaaiiOaiaadIhacaGG+bGaamOzaiaacIcacaWG4bGaai4l aiabeI7aXjaacMcaaaa@40EF@  where f(x/θ)= x 2 θ 3 ( 1+θ ) x+1 ( 2+θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaamOzaiaacIcacaWG4bGaai4laiabeI7aXjaacMcacqGH 9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiaadIhajuaGpaWaaWbaaSqabK qaGeaajugWa8qacaaIYaaaaKqzGeGaeqiUdexcfa4damaaCaaaleqa jeaibaqcLbmapeGaaG4maaaaaOWdaeaajuaGpeWaaeWaaOWdaeaaju gib8qacaaIXaGaey4kaSIaeqiUdehakiaawIcacaGLPaaajuaGpaWa aWbaaSqabKqaGeaajugWa8qacaWG4bGaey4kaSIaaGymaaaajuaGda qadaGcpaqaaKqzGeWdbiaaikdacqGHRaWkcqaH4oqCaOGaayjkaiaa wMcaaaaaaaa@5A96@  is the pmf of Area–biased Poisson exponential distribution with θ>0, then

I x ( θ )= 20 θ 4 8 θ 3 192 θ 2 309θ144 θ 4 ( 1+θ ) 2 ( 2+θ ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaamysaKqba+aadaWgaaqcbasaaKqzadWdbiaadIhaaSWd aeqaaKqba+qadaqadaGcpaqaaKqzGeWdbiabeI7aXbGccaGLOaGaay zkaaqcLbsacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiaaikdacaaI WaGaeqiUdexcfa4damaaCaaaleqajeaibaqcLbmapeGaaGinaaaaju gibiabgkHiTiaaiIdacqaH4oqCjuaGpaWaaWbaaSqabKqaGeaajugW a8qacaaIZaaaaKqzGeGaeyOeI0IaaGymaiaaiMdacaaIYaGaeqiUde xcfa4damaaCaaaleqajeaibaqcLbmapeGaaGOmaaaajugibiabgkHi TiaaiodacaaIWaGaaGyoaiabeI7aXjabgkHiTiaaigdacaaI0aGaaG inaaGcpaqaaKqzGeWdbiabeI7aXLqba+aadaahaaWcbeqcbasaaKqz adWdbiaaisdaaaqcfa4aaeWaaOWdaeaajugib8qacaaIXaGaey4kaS IaeqiUdehakiaawIcacaGLPaaajuaGpaWaaWbaaSqabKqaGeaajugW a8qacaaIYaaaaKqbaoaabmaak8aabaqcLbsapeGaaGOmaiabgUcaRi abeI7aXbGccaGLOaGaayzkaaqcfa4damaaCaaaleqajeaibaqcLbma peGaaG4maaaaaaaaaa@7934@

is the Fisher Information Matrix of ABPED, with

Estimation of parameters

Method of moment (MOM) estimate: let x1, x2,x3…….xn be a sample of size n from ABPED (1.6) then equating the population parameter to the sample mean we obtain the MOM estimate θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbeI 7aXzaaiaaaaa@39C8@  of < θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaeqiUdehaaa@39D9@ of ABPED as

θ ˜ = ( x ¯ 3 )+ ( x ¯ 3 ) 2 +6( x ¯ 1 ) ( x ¯ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbeI 7aXzaaiaaeaaaaaaaaa8qacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWd biabgkHiTKqbaoaabmaak8aabaqcLbsapeGabmiEayaaraGaeyOeI0 IaaG4maaGccaGLOaGaayzkaaqcLbsacqGHRaWkjuaGdaGcaaGcpaqa aKqba+qadaqadaGcpaqaaKqzGeWdbiqadIhagaqeaiabgkHiTiaaio daaOGaayjkaiaawMcaaKqba+aadaahaaWcbeqcbasaaKqzadWdbiaa ikdaaaqcLbsacqGHRaWkcaaI2aqcfa4aaeWaaOWdaeaajugib8qace WG4bGbaebacqGHsislcaaIXaaakiaawIcacaGLPaaaaSqabaaak8aa baqcfa4dbmaabmaak8aabaqcLbsapeGabmiEayaaraGaeyOeI0IaaG ymaaGccaGLOaGaayzkaaaaaaaa@5B17@

Where x is the sample mean.

Maximum likelihood estimate (MLE): let x1, x2, x3…….xn be a sample of size n from ABPED (1.6) , the MLE estimate θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbeI 7aXzaaiaaaaa@39C8@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaeqiUdehaaa@39D9@  is obtained as

θ ˜ = ( x ¯ 3 )+ ( x ¯ 3 ) 2 +6( x ¯ 1 ) ( x ¯ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbeI 7aXzaaiaGcqaaaaaaaaaWdbiabg2da9maalaaapaqaa8qacqGHsisl daqadaWdaeaajugib8qaceWG4bGbaebakiabgkHiTiaaiodaaiaawI cacaGLPaaacqGHRaWkdaGcaaWdaeaapeWaaeWaa8aabaqcLbsapeGa bmiEayaaraGccqGHsislcaaIZaaacaGLOaGaayzkaaWdamaaCaaale qabaWdbiaaikdaaaGccqGHRaWkcaaI2aWaaeWaa8aabaqcLbsapeGa bmiEayaaraGccqGHsislcaaIXaaacaGLOaGaayzkaaaaleqaaaGcpa qaa8qadaqadaWdaeaajugib8qaceWG4bGbaebakiabgkHiTiaaigda aiaawIcacaGLPaaaaaaaaa@5408@

Goodness of fit for ABPED

The Area–biased Poisson exponential distribution has been fitted to a number of countable datasets, and compared with size–biased Poisson lindley distribution. The following examples are used to illustrate a few situations generating the Area–biased distribution and–their applications. Microsoft Excel has been used to facilitate the use of the size–biased models to real life data. The MOM and MLE estimates are used to fit the distributions and is presented in the tables below. The datasets include number of observations of size distributions i:e small groups in various public situations reported by James,17 Coleman and James,18 and Simnoff,19 thunderstorm datasets reported by Shankar et al.,16 for all these datasets the ABPED distribution gives much closer fit than SBPLD (Tables 3–6).

Size of groups

Observed frequency

Expected frequency

SBPLD

ABPED

1

1486

1532.5

1480.381

2

694

630.6

704.9073

3

195

191.9

188.8048

4

37

51.3

39.95664

5

10

12.8

7.43203

6

1

3.9

1.273997

TOTAL

2423

2423

2422.756

Estimation of parameters

 

13.766

1.2166

d.f

3

2

p-value

<0.01

0.74903

AICc

3.2732

2.99996

BIC

2.0649

1.7918

Table 3 Counts of group of people in public Places on a spring afternoon in Portland

Size of groups

Observed frequency

Expected frequency

SBPLD

ABPED

1

316

323

312.1584

2

141

132.5

148.1144

3

44

40.2

39.53136

4

5

10.7

8.336454

5

4

3.6

1.545125

Total

510

510

Estimation of parameters

 

3.021

0.9728

d.f

2

2

p-value

0.4

0.6148

AICc

3.51453

3.33331

BIC

1.79064

1.60942

Table 4 Counts of shopping groups-Eugene, spring, department store and public market

Size of groups

Observed frequency

Expected frequency

SBPLD

ABPED

1

305

314.4

302.7967

2

144

134.4

150.5768

3

50

42.5

42.1199

4

5

11.8

9.309185

5

2

3.1

1.808334

6

1

0.8

0.323734

Total

507

507

506.9

Estimation of parameters

 

6.415

2.8126

d.f

2

2

p-value

0.043

0.2451

AICc

3.2809

3.000006

BIC

2.0727

1.79177

Table 5 counts of play Groups-Eugene, spring, public playground D

X

Fo

Expected frequency

SBPD

ABPED

1

87

83.2

83.95544

2

25

30.5

29.27444

3

5

5.6

5.741838

4

3

0.7

0.889832

Total

120

120

119.8615

Estimation of parameters

 

1.624

1.0167

d.f

1

1

p-value

0.2025

0.3133

Table 6 Frequency of thunderstorm events containing X thunderstorms at cape kennedy for May

Conclusions

Area–biased Poisson exponential distribution (ABPED) has been derived by area biasing the Discrete Poisson exponential distribution (PED) introduced by Fazal & Bashir.1 The discussion on estimation and applications of the area–biased distribution demonstrates that ABPED has a practical use to real life data. Form AIC and BIC measures the proposed Area–biased model appears to offer substantial improvements in fit over the size–biased poisson Lindley model. Also the fitting in these tables reveal that the Area–biased distribution provides us with better fits in the situations where zero–class is missing gives a better fit than size–biased Poisson Lindley distribution (SBPLD) and size–biased Poisson distribution(SBPD) therefore we conclude that area–biased poisson exponential distribution is a good replacement of (SBPLD).

Acknowledgements

The author expresses thankfulness to the learned referee for her valuable suggestions which improved the quality of the paper.

Conflict of interests

None.

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