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eISSN: 2378-315X

Biometrics & Biostatistics International Journal

Research Article Volume 12 Issue 3

Weighted Adya distribution with properties and application

Rama Shanker,1 Kamlesh Kumar Shukla2

1Department of Statistics, Assam University, Silchar, Assam, India
2Department of Community Medicine, Noida International Institute of Medical Science, India

Correspondence: Kamlesh Kumar Shukla, Department of Community Medicine, Noida International Institute of Medical Science, India

Received: May 15, 2023 | Published: May 31, 2023

Citation: Shanker R, Shukla KK. Weighted Adya distribution with properties and application. Biom Biostat Int J. 2023;12(3):68-74. DOI: 10.15406/bbij.2023.12.00386

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Abstract

In this paper a weighted version of Adya distribution which includes Adya distribution has been suggested for modeling lifetime data. The natures of descriptive statistics including coefficients of variation, skewness, kurtosis, and index of dispersion have been studied. The reliability measures including hazard rate function, reversed hazard rate function, mean residual life function and stochastic ordering have been studied. Method of maximum likelihood estimation has been discussed for estimating the parameters. A simulation study has been presented to know the performance of maximum likelihood estimates of parameters. The goodness of fit of the proposed distribution has been explained with a real lifetime data.

Keywords: Adya distribution, moments, reliability properties, maximum likelihood estimation, application

Introduction

As we know that the basic purpose of distribution theory is to determine a reasonable distributional model for the data arising from different fields of knowledge. And once the distributional model of the data is determined, characterization of distribution, computation of confidence intervals of parameters and critical regions for hypothesis tests can easily be done. It has been observed that the discrete or continuous data that we are getting are stochastic in nature and the existing distributions are not able to explain the true nature of data and this is the reasons for the search for new distributions. Further, it has been observed that many times the true nature of data can be better explained by weighted distribution with appropriate weight function. The concept of weighted distributions was firstly introduced by Fisher1 to model ascertainment biases which were later reformulated by Rao2 in a unifying theory for problems where the observations fall in the category of non-experimental, non-replicated and non-random. When observations are recorded by an investigator in the nature according to some stochastic model, the distribution of the recorded observations will not have the original distribution unless every observation is given an equal chance of being recorded. For instance, let the original observation x0  comes from a distribution having probability density function (pdf) f(x,θ) , where θ  may be a parameter vector and the observation x is recorded according to a probability re-weighted by weight function w(x,α)>0 , α  being a new parameter vector, then x comes from a distribution having pdf

fw(x;θ,α)=w(x;α)f(x;θ)E(W(X,α))   (1.1)

Note that such types of distribution are known as weighted distributions. The weighted distributions with weight function w(x,α)=x are called length biased distributions or simple size-biased distributions. Patil et al.3,4 have examined some general probability models leading to weighted probability distributions and their applications and showed the occurrence of w(x;α)=x  in a natural way in problems relating to sampling.

The study of weighted distributions is useful in distribution theory because it provides a new understanding of the existing standard probability distributions and it provides methods for extending existing standard probability distributions for modeling lifetime data due to introduction of additional parameter in the model which creates flexibility in their behavior. Weighted distributions occur in modeling datasets having clustered sampling, heterogeneity, and extraneous variation.

Shanker et al.5 introduced a one parameter lifetime distribution named Adya distribution (AD) having pdf and cdf

f1(x;θ)=θ3θ4+2θ2+2(θ+x)2eθx;x>0,θ>0   (1.2)

F1(x,θ)=1[1+θx(θx+2θ2+2)θ4+2θ2+2]eθx;x>0,θ>0   (1.3)

Obviously Adya distribution is a convex combination of an exponential (θ) distribution, a gamma (2,θ) distribution and a gamma (3,θ) distribution with their mixing proportions θ4θ4+2θ2+2 , 2θ2θ4+2θ2+2 and 2θ4+2θ2+2 , respectively.

Shanker et al.5 discussed its statistical properties including coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviations from the mean and the median, Bonferroni and Lorenz curves, stress-strength reliability along with estimation of parameter using maximum likelihood estimation and applications to model lifetime data from engineering and biomedical sciences.

The main purpose of the present paper is to derive weighted version of Adya distribution and discuss its important statistical properties. Its statistical properties including behaviour of pdf, cdf, hazard rate function, mean residual life function and moments based descriptive measures. Maximum likelihood estimation has been discussed to estimate parameters of the distribution. The simulation study to know the performance of maximum likelihood estimates is presented. Finally, an application of the proposed weighted Adya distribution has been presented to test its goodness of fit with other distributions.

Weighted Adya distribution

Considering the weight function w(x;α)=xα1 in (1.1) and using the pdf of Adya distribution from (1.2), the pdf of weighted Adya distribution (WAD) can be expressed as

f2(x;θ,α)=θα+2θ4+2θ2α+α(α+1)xα1Γ(α)(θ+x)2eθx;x>0,θ>0,α>0 , (2.1)

 where Γ(α) is the complete gamma function defined as

Γ(α)=0eyyα1dy;y>0,α>0   (2.2)

It can be easily shown that Adya distribution is a particular case of WAD at α=1 . Like Adya distribution, the pdf of Weighted Adya distribution can be easily expressed as a convex combination of gamma (θ,α) , gamma (θ,α+1)  and gamma (θ,α+2) distributions. We have

  f2(x;θ,α)=p1g1(x;θ,α)+p2g2(x;θ,α+1)+(1p1p2)g3(x;θ,α+2) , (2.3)

 where

 p1=θ4θ4+2θ2α+α(α+1) , p2=2θ2αθ4+2θ2α+α(α+1)

 g1(x;θ,α)=θαΓ(α)eθxxα1 , g2(x;θ,α+1)=θα+1Γ(α+1)eθxxα+11  and

  g3(x;θ,α+2)=θα+2Γ(α+2)eθxxα+21 .

The survival (reliability) function of WAD can be obtained as

S(x;θ,α)=P(X>x)=xf3(t;θ,α)dt=θα+2[θ4+2θ2α+α(α+1)]Γ(α)xtα1(θ2+2θt+t2)eθtdt  

  =[θ4+2θ2α+α(α+1)]Γ(α,θx)+(θx)α(θx+2θ2+α+1)eθx[θ4+2θ2α+α(α+1)]Γ(α) , (2.5)

 where Γ(α,z)  the upper incomplete gamma function defined as

Γ(α,z)=zeyyα1dy;y0,α>0   (2.6)

Thus, the cdf of WAD can thus be given by

F1(x;θ,α)=1S(x;θ,α)=1[θ4+2θ2α+α(α+1)]Γ(α,θx)+(θx)α(θx+2θ2+α+1)eθx[θ4+2θ2α+α(α+1)]Γ(α)

Graphs of the pdf and the cdf of WAD for varying values of the parameters θandα are shown in Figures 1 & 2 respectively.

Figure 1 pdf of WAD for varying values of θandα .

Figure 2 cdf of WAD for varying values of θandα .

Moments based measures

The r th moment about origin μr of WAD can be obtained as

μr=E(Xr)=Γ(α+r)Γ(α)θ4+2(α+r)θ2+(α+r)(α+r+1)θr{θ4+2θ2α+α(α+1)};r=1,2,3,...   (3.1)

The first four moments about origin of WAD thus can be obtained as

μ1=α{θ4+2(α+1)θ2+(α+1)(α+2)}θ{θ4+2θ2α+α(α+1)}  

μ2=α(α+1){θ4+2(α+2)θ2+(α+2)(α+3)}θ2{θ4+2θ2α+α(α+1)}  

μ3=α(α+1)(α+2){θ4+2(α+3)θ2+(α+3)(α+4)}θ3{θ4+2θ2α+α(α+1)}  

μ4=α(α+1)(α+2)(α+3){θ4+2(α+4)θ2+(α+4)(α+5)}θ4{θ4+2θ2α+α(α+1)}  .

Using the relationship between central moments and moments about origin, the central moments of WAD can be obtained as

μ2=α{θ8+(4α+4)θ6+(6α2+12α+6)θ4+(4α3+12α2+8α)θ2+(α4+4α3+5α2+2α)}θ2{θ2+2θ2α+α(α+1)}2  

μ3=2α{θ12+(6α+6)θ10+(15α2+27α+12)θ8+(20α3+50α2+30α)θ6+(15α4+48α3+39α2+6α)θ4+(6α5+24α4+30α3+12α2)θ2+(α6+5α5+9α4+7α3+2α2)}θ3{θ2+2θ2α+α(α+1)}3  

μ4=3α{(α+2)θ16+(8α2+24α+16)θ14+(28α3+112α2+124α+40)θ12+(56α4+280α3+416α2+192α)θ10+(70α5+420α4+782α3+488α2+56α)θ8+(56α6+392α5+888α4+760α3+208α2)θ6+(28α7+224α6+608α5+696α4+324α3+40α2)θ2+(8α8+72α7+232α6+344α5+240α4+64α3)θ+(α9+10α8+38α7+72α6+73α5+38α4+8α3)}θ4{θ2+2θα+α(α+1)}4

Thus the coefficient of variation (C.V), coefficient of skewness (β1) , coefficient of kurtosis (β2) , and index of dispersion (γ) of WAD are obtained as

C.V.=σμ1=θ8+(4α+4)θ6+(6α2+12α+6)θ4+(4α3+12α2+8α)θ2+(α4+4α3+5α2+2α)α{θ4+2(α+1)θ2+(α+1)(α+2)}  

β1=μ3μ23/2=2{θ12+(6α+6)θ10+(15α2+27α+12)θ8+(20α3+50α2+30α)θ6+(15α4+48α3+39α2+6α)θ4+(6α5+24α4+30α3+12α2)θ2+(α6+5α5+9α4+7α3+2α2)}α{θ8+(4α+4)θ6+(6α2+12α+6)θ4+(4α3+12α2+8α)θ2+(α4+4α3+5α2+2α)}3/2  

β2=μ4μ22=3{(α+2)θ16+(8α2+24α+16)θ14+(28α3+112α2+124α+40)θ12+(56α4+280α3+416α2+192α)θ10+(70α5+420α4+782α3+488α2+56α)θ8+(56α6+392α5+888α4+760α3+208α2)θ6+(28α7+224α6+608α5+696α4+324α3+40α2)θ2+(8α8+72α7+232α6+344α5+240α4+64α3)θ+(α9+10α8+38α7+72α6+73α5+38α4+8α3)}α{θ8+(4α+4)θ6+(6α2+12α+6)θ4+(4α3+12α2+8α)θ2+(α4+4α3+5α2+2α)}2  

γ=σ2μ1={θ8+(4α+4)θ6+(6α2+12α+6)θ4+(4α3+12α2+8α)θ2+(α4+4α3+5α2+2α)}θ{θ4+2θ2α+α(α+1)}{θ4+2(α+1)θ2+(α+1)(α+2)} .

It should be noted that these moments of WAD reduce to the corresponding moments of Adya distribution at α=1 . Behaviors of coefficient of variation (C.V), coefficient of Skewness (S.K), coefficient of kurtosis (S.K.) and index of dispersion (I.D) of WAD are drawn for varying values of parameters  and are shown in Figures 3–6 respectively.

Figure 3Graphs of C.V of WAD for varying values of parameters θandα .

Figure 4 Graphs of C.S of WAD for varying values of parameters θandα .

Figure 5 Graphs of C.K of WAD for varying values of parameters θandα .

Figure 6 Graphs of I.D of WAD for varying values of parameters θandα .

Moment generating function

The moment generating function of WAD can be obtained as

MX(t)=j=0tjj!μj  .

=j=0tjj!Γ(α+j)Γ(α)θ4+2(α+j)θ2+(α+j)(α+j+1)θj{θ4+2θ2α+α(α+1)}  .

=1{θ4+2θ2α+α(α+1)}Γ(α)j=0tjj!Γ(α+j){θ4+2(α+j)θ2+(α+j)(α+j+1)}θj

It can be easily verified that the coefficient of tjj!  in MX(t) gives the same μr as given by (3.1).

Harmonic mean

The harmonic mean of WAD can be obtained as

HM=E(1X)=θα+2{θ4+2θ2α+α(α+1)}Γ(α)01xxα1(θ+x)2eθxdx  

=θα+2{θ4+2θ2α+α(α+1)}Γ(α)0xα2(θ2+2θx+x2)eθxdx   

=θα+2{θ4+2θ2α+α(α+1)}Γ(α)[θ20eθxxα11dx+2θ0eθxxα1dx+0eθxxα+11dx]  

=θα+2{θ4+2θ2α+α(α+1)}Γ(α)[θ2Γ(α1)θα1+2θΓ(α)θα+Γ(α+1)θα+1]

  =θ{θ4Γ(α1)2θ2Γ(α)+Γ(α+1)}{θ4+2θ2α+α(α+1)}Γ(α);α1 .

Reliability measures

There are some important reliability measures of a distribution namely, the hazard rate function, reverse hazard rate function, Mills ratio and inverse Mills ratio, the mean residual life function and Stochastic ordering. In this section these reliability measures for WAD have been discussed.

Hazard rate function

The hazard (or instantaneous failure rate function) plays a crucial role in reliability and survival analysis, as it defines the conditional probability of failure of an item in the next very small units of time Δx , given that it did not fail before x . Suppose X is a random variable with cdf F(x)=P(Xx) . If F(x)  is absolutely continuous, then the random variable X  has a probability density function f(x) . The hazard rate (HR) function h(x)  of X  is defined as

h(x)=f(x)1F(x)=f(x)S(x)   

The hazard (or failure rate) function, h(x) of WAD can be obtained as

h(x;θ,α)=f3(x;θ,α)S(x;θ,α)=θα+2xα1(θ+x)2eθx{θ4+2θ2α+α(α+1)}Γ(α,θx)+(θx)α(θx+2θ2+α+1)eθx  .

Graphs of h(x) of WAD for varying values of parameters θandα are shown in Figure 7. It is obvious that for varying values of parameters, the shapes of hazard rate function of WAD are changing and it can be used for data of various nature.

Figure 7 h(x) of WAD for varying values of parameters θandα .

Reverse hazard rate function

A function closely related to the hazard rate function is the reverse hazard rate function which was firstly introduced by Barlow et al. It is the dual of the hazard rate function and is defined as

r(x)=f(x)F(x)  .

Thus, the corresponding reverse hazard rate function of WAD can be obtained as

 r(x)=θα+2xα1(θ+x)2eθx{θ4+2θ2α+α(α+1)}Γ(α)[{θ4+2θ2α+α(α+1)}Γ(α,θx)+(θx)α(θx+2θ2+α+1)eθx] .

Mills ratio and inverse mills ratio

 The Mills ratio is defined as the ratio of the complementary cdf to the pdf of a random variable X and is denoted as m(x)  and defined as

 m(x)=1F(x)f(x)=S(x)f(x)={θ4+2θ2α+α(α+1)}Γ(α,θx)+(θx)α(θx+2θ2+α+1)eθxθα+2xα1(θ+x)2eθx .

It is also related to the hazard rate function as

m(x)=1h(x)  

The inverse Mills ratio is the ratio of the pdf to the complementary cdf of a random variable X .

Mean residual life function

The mean residual life function m(x)=E(Xx|X>x)  of WAD can be obtained as

m(x;θ,α)=1S(x;θ,α)xtf1(t;θ,α)dtx  

=1S(x;θ,α)[θα+2{θ4+2θ2α+α(α+1)}Γ(α)xtα(θ+2θt+t2)eθtdt]x  

                                  =(θx)α[θx+θ4+2(α+1)θ2+(α+1)(α+2)]eθx+[αθ4+α(α+1)(2θ2+α+2)θx{θ4+2θ2α+α(α+1)}]Γ(α,θx)θ[(θx)α(θx+2θ2+α+1)eθx+{θ4+2θ2α+α(α+1)}Γ(α,θx)]

It can be easily shown that m(0;θ,α)=α{θ4+2(α+1)θ2+(α+1)(α+2)}θ{θ4+2θ2α+α(α+1)}=μ1 .

Graphs of m(x) of WAD for varying values of parameters θandα  are shown in Figure 8.

Figure 8 m(x) of WAD for varying values of parameters θandα .

Stochastic ordering

The stochastic ordering of positive continuous random variables is an important tool for judging their comparative behavior. A random variable X is said to be smaller than a random variable Y in the

  1. stochastic order (XstY) if FX(x)FY(x) for all x
  2. hazard rate order (XhrY) if hX(x)hY(x) for all x
  3. mean residual life order (XmrlY) if mX(x)mY(x) for all x
  4. Likelihood ratio order (XlrY) if fX(x)fY(x) decreases in x .

The following interrelationships due to Shaked & Shanthikumar6 are well known for establishing stochastic ordering of distributions

XlrYXhrYXmrlY   

XstY  

It can be easily shown that WAD is ordered with respect to the strongest ‘likelihood ratio’ ordering. The stochastic ordering of WAD has been explained in the following theorem:

Theorem: Suppose X WAD (θ1,α1)  and Y  WAD (θ2,α2) . If α1α2andθ1>θ2  (or α1<α2 ; θ1θ2  ), then XlrY and hence XhrY ,XmrlY and XstY .

Proof: We have

fX(x;θ1,α1)fY(x;θ2,α2)=[θ1α1+2{θ24+2θ22α2+α2(α2+1)}Γ(α2)θ2α2+2{θ14+2θ12α1+α1(α1+1)}Γ(α1)](θ1+xθ2+x)2xα1α2e(θ1θ2)x;x>0   

Now, taking logarithm both sides, we get

logfX(x;θ1,α1)fY(x;θ2,α2)=log[θ1α1+2{θ24+2θ22α2+α2(α2+1)}Γ(α2)θ2α2+2{θ14+2θ12α1+α1(α1+1)}Γ(α1)]+2log(θ1+xθ2+x)+(α1α2)logx(θ1θ2)x  

 This gives ddxlnfX(x;θ1,α1)fY(x;θ2,α2)=2(θ2θ1)(θ1+x)(θ2+x)+α1α2x(θ1θ2)  .

Thus for α1α2andθ1>θ2  (or α1<α2  and θ1θ2  ), ddxlnfX(x;θ1,α1)fY(x;θ2,α2)<0 . This means that XlrY and hence XhrY , XmrlY and XstY .

Maximum likelihood estimation

Suppose (x1,x2,x3,...,xn)  be a random sample of size n  from WAD. The log-likelihood function,logL of WAD can be obtained as

logL=ni=1logf2(xi;θ,α)  

=n[(α+2)logθlog(θ4+2θ2α+α2+α)logΓ(α)]+(α1)ni=1log(xi)+2ni=1log(θ+xi)nθˉx  

The maximum likelihood estimates (MLE’s) (ˆθ,ˆα)  of the parameters (θ,α)  of WAD are the solutions of the following log likelihood equations

logLθ=n(α+2)θ4n(θ3+θα)θ4+2θ2α+α2+α+2ni=11θ+xinˉx=0   

logLα=nlnθn(2θ2+2α+1)θ4+2θ2α+α2+αnψ(α)+ni=1log(xi)=0   

 where ψ(α)=ddαlogΓ(α) is the digamma function.

Since these two log likelihood equations cannot be expressed in closed forms, they cannot be solved analytically. However, these two log likelihood equations can be solved using R-software.

A simulation study

In this section, a simulation study has been carried to check the performance of maximum likelihood estimates by taking sample sizes (n=20, 40, 60, 80) for values of θ=0.5,1.0,1.5,2.0  and α=0.5  & 1.5 . Acceptance and rejection method is used to generate random number for data simulation using R-software. The process was repeated 1,000 times for the calculation of Average Bias error (ABE) and MSE (Mean square error) of parameters θ and α  and are presented in Tables1 & 2 respectively. For the WAD decreasing trend has been observed in ABE and MSE as the sample size increases and this shows that the performance of maximum likelihood estimators is quite good and consistent.

n

θ  

ABE(θ )

MSE (θ )

ABE (α )

MSE (α )

20

0.5

-0.0037

0.0003

-0.0560

0.0007

20

1.0

-0.0287

0.01649

-0.0310

0.0193

20

1.5

-0.0537

0.0577

-0.0560

0.0628

20

2.0

-0.0787

0.1239

-0.0810

0.1313

40

0.5

0.0022

0.0002

-0.0225

0.0003

40

1.0

-0.0103

0.0042

-0.0099

0.0039

40

1.5

-0.0227

0.0207

-0.02249

0.0202

40

2.0

-0.0353

0.0497

-0.0349

0.0489

60

0.5

0.001

0.0006

-0.0142

0.0003

60

1.0

-0.0073

0.0032

-0.0059

0.0021

60

1.5

-0.0156

0.0147

-0.0142

0.0121

60

2.0

-0.0239

0.0345

-0.0225

0.0305

80

0.5

0.0001

0.0000

-0.0117

0.0004

80

1.0

-0.0061

0.0030

-0.0055

0.0024

80

1.5

-0.0124

0.0123

-0.0117

0.0111

80

2.0

-0.0186

0.0277

-0.0180

0.0360

Table 1 ABE and MSE of parameters at fixed value α=0.5

n

 θ

  ABE(θ )

MSE (θ )

ABE (α )

 MSE (α )

20

0.5

  0.0298

0.0177

 0.0052

0.0608

20

1.0

  0.0048

0.0005

 0.0302

0.0182

20

1.5

 -0.0202

0.0081

 0.0052

0.0005

20

2.0

 -0.0452

0.0408

-0.0198

0.0078

40

0.5

 0.0166

0.0112

 0.0162

0.0678

40

1.0

 0.0041

0.0007

 0.0287

0.0329

40

1.5

 -0.0084

0.0028

 0.0162

0.0104

40

2.0

 -0.0208

0.0174

 0.0036

0.0005

60

0.5

 0.0084

0.0094

 0.0067

0.0329

60

1.0

 0.0001

0.0000

 0.0151

0.0136

60

1.5

 -0.0082

0.0040

 0.0067

0.0027

60

2.0

 -0.0165

0.0164

 -0.0015

0.0001

80

0.5

 0.0070

0.0039

 0.0066

0.0293

80

1.0

 0.0007

0.0004

 0.0129

0.0133

80

1.5

 -0.0055

0.0024

 0.0066

0.0035

80

2.0

 -0.0117

0.0109

 0.0004

0.0001

Table 2 ABE and MSE of parameters at fixed value α=0.5

A numerical example

In this section application and the goodness of fit of WAD has been discussed with the following real lifetime data relating to the lifetime of a certain device reported by Sylwia7 and the observations are 0.0094, 0.0500, 0.4064, 4.6307, 5.1741, 5.8808, 6.3348, 7.1645, 7.2316, 8.2604, 9.2662, 9.3812, 9.5223, 9.8783, 9.9346, 10.0192, 10.4077, 10.4791, 11.0760, 11.3250, 11.5284, 11.9226, 12.0294, 12.0740, 12.1835, 12.3549, 12.5381, 12.8049, 13.4615, 13.8530.

For this data set, WAD has been fitted along with one parameter exponential and Adya distributions and two-parameter distributions including weighted Lindley distribution (WLD) introduced by Ghitany et al.,8 quasi Lindley distribution (QLD) of Shanker & Mishra,9 a two-parameter Lindley distribution (TPLD-I) of Shanker & Mishra,10 a two-parameter Lindley distribution (TPLD-II) of Shanker et al.11 and Weibull distribution introduced by Weibull.12 The pdf and cdf of the considered distributions: WLD, Weibull, TPLD-I, TPLD-II, QLD and exponential distribution are given in Table 3. The ML estimates, values of 2lnL , Akaike Information criteria (AIC), K-S statistics and p-value of the fitted distributions are presented in Table 4. The AIC and K-S Statistics are computed using the following formulae: AIC=2lnL+2k  and K-S=Supx|Fn(x)F0(x)| , where k  = the number of parameters, n  = the sample size , Fn(x) is the empirical (sample) cumulative distribution function, and F0(x)  is the theoretical cumulative distribution function. The best distribution is the distribution corresponding to lower values of 2lnL , AIC, and K-S statistics.

Distributions

Pdf

Cdf

WLD

f(x;θ,α)=θα+1θ+αxα1Γ(α+1)(1+x)eθx   F(x;θ,α)=1(θ+α)Γ(α,θx)+(θx)αeθx(θ+α)Γ(α)  

Weibull

f(x;θ,α)=θαxα1eθxα   F(x;θ,α)=1eθxα  

TPLD-I

f(x;θ,α)=θ2θα+1(α+x)eθx;x>0,θ>0,θα>1   F(x;θ,α)=1(1+θxαθ+1)eθx  

TPLD-II

f(x;θ,α)=θ2θ+α(1+αx)eθx ;x>0,θ>0,α>0   F(x;θ,α)=1(1+αθxθ+α)eθx  

QLD

F(x;θ,α)=1(1+αθxθ+α)eθx   F(x;θ,α)=1(1+θxα+1)eθx  

Exponential (ED)

f(x;θ)=θeθx   F(x;θ)=1eθx  

Table 3 The pdf and the cdf of fitted distributions

Distribution

ML Estimates

2lnL  

AIC

K-S

P-value

θ

α

WAD

0.2303

0.132

179.31

183.31

0.276

0.010

WLD

0.1733

0.7537

183

187

0.281

0.012

Weibull

0.0258

1.6168

185.45

189.45

0.278

0.014

TPLD-I

0.2001

1.1774

183.74

187.74

0.275

0.016

TPLD-II

0.2002

0.849

183.74

187.74

0.275

0.016

QLD

0.2001

0.2356

183.74

187.74

0.275

0.016

AD

0.3314

-------

187.43

189.43

0.338

0.001

Exponential

0.1106

--------

192.09

194.09

0.314

0.004

Table 4 MLE’s, - 2ln L, AIC, K-S statistics and p-values of the fitted distributions

 

The variance-covariance matrix and the 95% confidence intervals (CI’s) of the ML estimates of the parameters θ  and α of WAD are presented in Table 5. From the goodness of fit in Table 4 and the fitted plots of the distribution for the considered datasets in Figure 9 shows that WAD gives much closure fit as compared to other considered distributions.

Parameters

Variance-covariance matrix

95 % CI

 

 ˆθ                                         ˆα

Lower                    Upper

ˆθ  

0.001109402             0.00279572

0.17450148         0.3121785

ˆα  

0.002795720             0.01862830

0.01988068         0.6389076

Table 5 Variance-Covariance matrix and 95% confidence intervals (CI’s) of the ME estimates (ˆθ,ˆα) of parameters (θ,α) for the given dataset

 

The profile of likelihood estimates of parameters ˆθandˆα  of WAD for the given dataset is shown in Figure 9. Also, the fitted plots of the considered dataset for WAD are shown in Figure 10.

Figure 9 Profile of the likelihood estimates θandα of WAD for the given dataset.

Figure 10 Fitted plots of the two-parameter distributions for the given dataset.

Concluding remarks

In this paper a two-parameter weighted Adya distribution (WAD) which includes one parameter Adya distribution as special case has been suggested for modeling lifetime data. Some of its properties including shapes of probability density function, moments-based measures including coefficients of variation, skewness, kurtosis, and index of dispersion; hazard rate function, mean residual life function and stochastic ordering have been studied. Method of maximum likelihood estimation has been discussed for estimating the parameters. The simulation study has been presented to know the performance of parameters. The goodness of the proposed distribution has been explained with a real lifetime data and the fit has been found to be quite satisfactory over one parameter and two parameter lifetime distributions.

Acknowledgments

None.

Conflicts of interest

The authors declare that there are no conflicts of interest.

Funding

None.

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