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

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 $x_0$  comes from a distribution having probability density function (pdf) $f\left(x,\theta \right)$ , where $\theta$  may be a parameter vector and the observation $x$ is recorded according to a probability re-weighted by weight function $w\left(x,\alpha \right)>0$ , $\alpha$  being a new parameter vector, then $x$ comes from a distribution having pdf

$f_w\left(x;\theta ,\alpha \right)=\frac{w\left(x;\alpha \right)f\left(x;\theta \right)}{E\left(W\left(X,\alpha \right)\right)}$   (1.1)

Note that such types of distribution are known as weighted distributions. The weighted distributions with weight function $w\left(x,\alpha \right)=x$ 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 $w\left(x;\alpha \right)=x$  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

$f_1\left(x;\theta \right)=\frac{{\theta }^3}{{\theta }^4+2{\theta }^2+2}{\left(\theta +x\right)}^2{e}^{-\theta x};x>0,\theta >0$   (1.2)

$F_1\left(x,\theta \right)=1-\left[1+\frac{\theta x\left(\theta x+2{\theta }^2+2\right)}{{\theta }^4+2{\theta }^2+2}\right]e^{-\theta x};x>0,\theta >0$   (1.3)

Obviously Adya distribution is a convex combination of an exponential $\left(\theta \right)$ distribution, a gamma $\left(2,\theta \right)$ distribution and a gamma $\left(3,\theta \right)$ distribution with their mixing proportions $\frac{{\theta }^4}{{\theta }^4+2{\theta }^2+2}$ , $\frac{2{\theta }^2}{{\theta }^4+2{\theta }^2+2}$ and $\frac{2}{{\theta }^4+2{\theta }^2+2}$ , 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.

Considering the weight function $w\left(x;\alpha \right)=x^{\alpha -1}$ in (1.1) and using the pdf of Adya distribution from (1.2), the pdf of weighted Adya distribution (WAD) can be expressed as

$f_2\left(x;\theta ,\alpha \right)=\frac{{\theta }^{\alpha +2}}{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)}\frac{x^{\alpha -1}}{\Gamma \left(\alpha \right)}{\left(\theta +x\right)}^2{e}^{-\theta x};x>0,\theta >0,\alpha >0$ , (2.1)

 where $\Gamma \left(\alpha \right)$ is the complete gamma function defined as

$\Gamma \left(\alpha \right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-y}}{y}^{\alpha -1}dy;y>0,\alpha >0$   (2.2)

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

  $f_2\left(x;\theta ,\alpha \right)=p_1{g}_1\left(x;\theta ,\alpha \right)+p_2{g}_2\left(x;\theta ,\alpha +1\right)+\left(1-p_1-p_2\right)g_3\left(x;\theta ,\alpha +2\right)$ , (2.3)

 where

 $p_1=\frac{{\theta }^4}{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)}$ , $p_2=\frac{2{\theta }^2\alpha }{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)}$

 $g_1\left(x;\theta ,\alpha \right)=\frac{{\theta }^{\alpha }}{\Gamma \left(\alpha \right)}{e}^{-\theta x}{x}^{\alpha -1}$ , $g_2\left(x;\theta ,\alpha +1\right)=\frac{{\theta }^{\alpha +1}}{\Gamma \left(\alpha +1\right)}{e}^{-\theta x}{x}^{\alpha +1-1}$  and

  $g_3\left(x;\theta ,\alpha +2\right)=\frac{{\theta }^{\alpha +2}}{\Gamma \left(\alpha +2\right)}{e}^{-\theta x}{x}^{\alpha +2-1}$ .

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

$S\left(x;\theta ,\alpha \right)=P\left(X>x\right)={\displaystyle \underset{x}{\overset{\infty }{\int }}{f}_3\left(t;\theta ,\alpha \right)}dt=\frac{{\theta }^{\alpha +2}}{\left[{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right]\Gamma \left(\alpha \right)}{\displaystyle \underset{x}{\overset{\infty }{\int }}{t}^{\alpha -1}}\left({\theta }^2+2\theta t+t^2\right)e^{-\theta t}dt$  

  $=\frac{\left[{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right]\Gamma \left(\alpha ,\theta x\right)+{\left(\theta x\right)}^{\alpha }\left(\theta x+2{\theta }^2+\alpha +1\right)e^{-\theta x}}{\left[{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right]\Gamma \left(\alpha \right)}$ , (2.5)

 where $\Gamma \left(\alpha ,z\right)$  the upper incomplete gamma function defined as

$\Gamma \left(\alpha ,z\right)={\displaystyle \underset{z}{\overset{\infty }{\int }}{e}^{-y}{y}^{\alpha -1}dy;y\ge 0,\alpha >0}$   (2.6)

Thus, the cdf of WAD can thus be given by

$\begin{array}{l}{F}_1\left(x;\theta ,\alpha \right)=1-S\left(x;\theta ,\alpha \right)\\ =1-\frac{\left[{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right]\Gamma \left(\alpha ,\theta x\right)+{\left(\theta x\right)}^{\alpha }\left(\theta x+2{\theta }^2+\alpha +1\right)e^{-\theta x}}{\left[{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right]\Gamma \left(\alpha \right)}\end{array}$

Graphs of the pdf and the cdf of WAD for varying values of the parameters $\theta \text{and}\alpha$ are shown in Figures 1 & 2 respectively.

Figure 1 pdf of WAD for varying values of $\theta \text{and}\alpha$ .

Figure 2 cdf of WAD for varying values of $\theta \text{and}\alpha$ .

The $r$ th moment about origin ${\mu }_r{}^{\prime }$ of WAD can be obtained as

${\mu }_r{}^{\prime }=E\left(X^r\right)=\frac{\Gamma \left(\alpha +r\right)}{\Gamma \left(\alpha \right)}\frac{{\theta }^4+2\left(\alpha +r\right){\theta }^2+\left(\alpha +r\right)\left(\alpha +r+1\right)}{{\theta }^r\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}};r=1,2,3,\mathrm{...}$   (3.1)

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

${\mu }_1{}^{\prime }=\frac{\alpha \left\{{\theta }^4+2\left(\alpha +1\right){\theta }^2+\left(\alpha +1\right)\left(\alpha +2\right)\right\}}{\theta \left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}}$  

${\mu }_2{}^{\prime }=\frac{\alpha \left(\alpha +1\right)\left\{{\theta }^4+2\left(\alpha +2\right){\theta }^2+\left(\alpha +2\right)\left(\alpha +3\right)\right\}}{{\theta }^2\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}}$  

${\mu }_3{}^{\prime }=\frac{\alpha \left(\alpha +1\right)\left(\alpha +2\right)\left\{{\theta }^4+2\left(\alpha +3\right){\theta }^2+\left(\alpha +3\right)\left(\alpha +4\right)\right\}}{{\theta }^3\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}}$  

${\mu }_4{}^{\prime }=\frac{\alpha \left(\alpha +1\right)\left(\alpha +2\right)\left(\alpha +3\right)\left\{{\theta }^4+2\left(\alpha +4\right){\theta }^2+\left(\alpha +4\right)\left(\alpha +5\right)\right\}}{{\theta }^4\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}}$  .

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

${\mu }_2=\frac{\alpha \left\{{\theta }^8+\left(4\alpha +4\right){\theta }^6+\left(6{\alpha }^2+12\alpha +6\right){\theta }^4+\left(4{\alpha }^3+12{\alpha }^2+8\alpha \right){\theta }^2+\left({\alpha }^4+4{\alpha }^3+5{\alpha }^2+2\alpha \right)\right\}}{{\theta }^2{\left\{{\theta }^2+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}}^2}$  

${\mu }_3=\frac{2\alpha \left\{\begin{array}{l}{\theta }^{12}+\left(6\alpha +6\right){\theta }^{10}+\left(15{\alpha }^2+27\alpha +12\right){\theta }^8+\left(20{\alpha }^3+50{\alpha }^2+30\alpha \right){\theta }^6\\ +\left(15{\alpha }^4+48{\alpha }^3+39{\alpha }^2+6\alpha \right){\theta }^4+\left(6{\alpha }^5+24{\alpha }^4+30{\alpha }^3+12{\alpha }^2\right){\theta }^2\\ +\left({\alpha }^6+5{\alpha }^5+9{\alpha }^4+7{\alpha }^3+2{\alpha }^2\right)\end{array}\right\}}{{\theta }^3{\left\{{\theta }^2+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}}^3}$  

${\mu }_4=\frac{3\alpha \left\{\begin{array}{l}\left(\alpha +2\right){\theta }^{16}+\left(8{\alpha }^2+24\alpha +16\right){\theta }^{14}+\left(28{\alpha }^3+112{\alpha }^2+124\alpha +40\right){\theta }^{12}\\ +\left(56{\alpha }^4+280{\alpha }^3+416{\alpha }^2+192\alpha \right){\theta }^{10}+\left(70{\alpha }^5+420{\alpha }^4+782{\alpha }^3+488{\alpha }^2+56\alpha \right){\theta }^8\\ +\left(56{\alpha }^6+392{\alpha }^5+888{\alpha }^4+760{\alpha }^3+208{\alpha }^2\right){\theta }^6\\ +\left(28{\alpha }^7+224{\alpha }^6+608{\alpha }^5+696{\alpha }^4+324{\alpha }^3+40{\alpha }^2\right){\theta }^2\\ +\left(8{\alpha }^8+72{\alpha }^7+232{\alpha }^6+344{\alpha }^5+240{\alpha }^4+64{\alpha }^3\right)\theta \\ +\left({\alpha }^9+10{\alpha }^8+38{\alpha }^7+72{\alpha }^6+73{\alpha }^5+38{\alpha }^4+8{\alpha }^3\right)\end{array}\right\}}{{\theta }^4{\left\{{\theta }^2+2\theta \alpha +\alpha \left(\alpha +1\right)\right\}}^4}$

Thus the coefficient of variation (C.V), coefficient of skewness $\left(\sqrt{{\beta }_1}\right)$ , coefficient of kurtosis $\left({\beta }_2\right)$ , and index of dispersion $\left(\gamma \right)$ of WAD are obtained as

$C.V.=\frac{\sigma }{{\mu }_1{}^{\prime }}=\frac{\sqrt{{\theta }^8+\left(4\alpha +4\right){\theta }^6+\left(6{\alpha }^2+12\alpha +6\right){\theta }^4+\left(4{\alpha }^3+12{\alpha }^2+8\alpha \right){\theta }^2+\left({\alpha }^4+4{\alpha }^3+5{\alpha }^2+2\alpha \right)}}{\sqrt{\alpha }\left\{{\theta }^4+2\left(\alpha +1\right){\theta }^2+\left(\alpha +1\right)\left(\alpha +2\right)\right\}}$  

$\sqrt{{\beta }_1}=\frac{{\mu }_3}{{\mu }_2{}^{3/2}}=\frac{2\left\{\begin{array}{l}{\theta }^{12}+\left(6\alpha +6\right){\theta }^{10}+\left(15{\alpha }^2+27\alpha +12\right){\theta }^8+\left(20{\alpha }^3+50{\alpha }^2+30\alpha \right){\theta }^6\\ +\left(15{\alpha }^4+48{\alpha }^3+39{\alpha }^2+6\alpha \right){\theta }^4+\left(6{\alpha }^5+24{\alpha }^4+30{\alpha }^3+12{\alpha }^2\right){\theta }^2\\ +\left({\alpha }^6+5{\alpha }^5+9{\alpha }^4+7{\alpha }^3+2{\alpha }^2\right)\end{array}\right\}}{\sqrt{\alpha }{\left\{{\theta }^8+\left(4\alpha +4\right){\theta }^6+\left(6{\alpha }^2+12\alpha +6\right){\theta }^4+\left(4{\alpha }^3+12{\alpha }^2+8\alpha \right){\theta }^2+\left({\alpha }^4+4{\alpha }^3+5{\alpha }^2+2\alpha \right)\right\}}^{3/2}}$  

${\beta }_2=\frac{{\mu }_4}{{\mu }_2{}^2}=\frac{3\left\{\begin{array}{l}\left(\alpha +2\right){\theta }^{16}+\left(8{\alpha }^2+24\alpha +16\right){\theta }^{14}+\left(28{\alpha }^3+112{\alpha }^2+124\alpha +40\right){\theta }^{12}\\ +\left(56{\alpha }^4+280{\alpha }^3+416{\alpha }^2+192\alpha \right){\theta }^{10}+\left(70{\alpha }^5+420{\alpha }^4+782{\alpha }^3+488{\alpha }^2+56\alpha \right){\theta }^8\\ +\left(56{\alpha }^6+392{\alpha }^5+888{\alpha }^4+760{\alpha }^3+208{\alpha }^2\right){\theta }^6\\ +\left(28{\alpha }^7+224{\alpha }^6+608{\alpha }^5+696{\alpha }^4+324{\alpha }^3+40{\alpha }^2\right){\theta }^2\\ +\left(8{\alpha }^8+72{\alpha }^7+232{\alpha }^6+344{\alpha }^5+240{\alpha }^4+64{\alpha }^3\right)\theta \\ +\left({\alpha }^9+10{\alpha }^8+38{\alpha }^7+72{\alpha }^6+73{\alpha }^5+38{\alpha }^4+8{\alpha }^3\right)\end{array}\right\}}{\alpha {\left\{{\theta }^8+\left(4\alpha +4\right){\theta }^6+\left(6{\alpha }^2+12\alpha +6\right){\theta }^4+\left(4{\alpha }^3+12{\alpha }^2+8\alpha \right){\theta }^2+\left({\alpha }^4+4{\alpha }^3+5{\alpha }^2+2\alpha \right)\right\}}^2}$  

$\gamma =\frac{{\sigma }^2}{{\mu }_1{}^{\prime }}=\frac{\left\{{\theta }^8+\left(4\alpha +4\right){\theta }^6+\left(6{\alpha }^2+12\alpha +6\right){\theta }^4+\left(4{\alpha }^3+12{\alpha }^2+8\alpha \right){\theta }^2+\left({\alpha }^4+4{\alpha }^3+5{\alpha }^2+2\alpha \right)\right\}}{\theta \left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}\left\{{\theta }^4+2\left(\alpha +1\right){\theta }^2+\left(\alpha +1\right)\left(\alpha +2\right)\right\}}$ .

It should be noted that these moments of WAD reduce to the corresponding moments of Adya distribution at $\alpha =1$ . 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 $\theta \text{and}\alpha$ .

Figure 4 Graphs of C.S of WAD for varying values of parameters $\theta \text{and}\alpha$ .

Figure 5 Graphs of C.K of WAD for varying values of parameters $\theta \text{and}\alpha$ .

Figure 6 Graphs of I.D of WAD for varying values of parameters $\theta \text{and}\alpha$ .

The moment generating function of WAD can be obtained as

$M_X\left(t\right)={\displaystyle \sum _{j=0}^{\infty }\frac{t^j}{j!}}{\mu }_j{}^{\prime }$  .

$={\displaystyle \sum _{j=0}^{\infty }\frac{t^j}{j!}}\frac{\Gamma \left(\alpha +j\right)}{\Gamma \left(\alpha \right)}\frac{{\theta }^4+2\left(\alpha +j\right){\theta }^2+\left(\alpha +j\right)\left(\alpha +j+1\right)}{{\theta }^j\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}}$  .

$=\frac{1}{\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}\Gamma \left(\alpha \right)}{\displaystyle \sum _{j=0}^{\infty }\frac{t^j}{j!}}\frac{\Gamma \left(\alpha +j\right)\left\{{\theta }^4+2\left(\alpha +j\right){\theta }^2+\left(\alpha +j\right)\left(\alpha +j+1\right)\right\}}{{\theta }^j}$

It can be easily verified that the coefficient of $\frac{t^j}{j!}$  in $M_X\left(t\right)$ gives the same ${\mu }_r{}^{\prime }$ as given by (3.1).

The harmonic mean of WAD can be obtained as

$HM=E\left(\frac{1}{X}\right)=\frac{{\theta }^{\alpha +2}}{\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}\Gamma \left(\alpha \right)}{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{1}{x}}{x}^{\alpha -1}{\left(\theta +x\right)}^2{e}^{-\theta x}dx$  

$=\frac{{\theta }^{\alpha +2}}{\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}\Gamma \left(\alpha \right)}{\displaystyle \underset{0}{\overset{\infty }{\int }}{x}^{\alpha -2}\left({\theta }^2+2\theta x+x^2\right)e^{-\theta x}dx}$   

$=\frac{{\theta }^{\alpha +2}}{\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}\Gamma \left(\alpha \right)}\left[{\theta }^2{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\theta x}{x}^{\alpha -1-1}}dx+2\theta {\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\theta x}{x}^{\alpha -1}}dx+{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\theta x}{x}^{\alpha +1-1}}dx\right]$  

$=\frac{{\theta }^{\alpha +2}}{\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}\Gamma \left(\alpha \right)}\left[{\theta }^2\frac{\Gamma \left(\alpha -1\right)}{{\theta }^{\alpha -1}}+2\theta \frac{\Gamma \left(\alpha \right)}{{\theta }^{\alpha }}+\frac{\Gamma \left(\alpha +1\right)}{{\theta }^{\alpha +1}}\right]$

  $=\frac{\theta \left\{{\theta }^4\Gamma \left(\alpha -1\right)2{\theta }^2\Gamma \left(\alpha \right)+\Gamma \left(\alpha +1\right)\right\}}{\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}\Gamma \left(\alpha \right)};\alpha \ge 1$ .

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 $\Delta x$ , given that it did not fail before $x$ . Suppose $X$ is a random variable with cdf $F\left(x\right)=P\left(X\le x\right)$ . If $F\left(x\right)$  is absolutely continuous, then the random variable $X$  has a probability density function $f\left(x\right)$ . The hazard rate (HR) function $h\left(x\right)$  of $X$  is defined as

$h\left(x\right)=\frac{f\left(x\right)}{1-F\left(x\right)}=\frac{f\left(x\right)}{S\left(x\right)}$   

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

$h\left(x;\theta ,\alpha \right)=\frac{f_3\left(x;\theta ,\alpha \right)}{S\left(x;\theta ,\alpha \right)}=\frac{{\theta }^{\alpha +2}{x}^{\alpha -1}{\left(\theta +x\right)}^2{e}^{-\theta x}}{\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}\Gamma \left(\alpha ,\theta x\right)+{\left(\theta x\right)}^{\alpha }\left(\theta x+2{\theta }^2+\alpha +1\right)e^{-\theta x}}$  .

Graphs of $h\left(x\right)$ of WAD for varying values of parameters $\theta \text{and}\alpha$ 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 $h\left(x\right)$ of WAD for varying values of parameters $\theta \text{and}\alpha$ .

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

$r\left(x\right)=\frac{f\left(x\right)}{F\left(x\right)}$  .

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

 $r\left(x\right)=\frac{{\theta }^{\alpha +2}{x}^{\alpha -1}{\left(\theta +x\right)}^2{e}^{-\theta x}}{\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}\Gamma \left(\alpha \right)-\left[\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}\Gamma \left(\alpha ,\theta x\right)+{\left(\theta x\right)}^{\alpha }\left(\theta x+2{\theta }^2+\alpha +1\right)e^{-\theta x}\right]}$ .

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 $m\left(x\right)$  and defined as

 $m\left(x\right)=\frac{1-F\left(x\right)}{f\left(x\right)}=\frac{S\left(x\right)}{f\left(x\right)}=\frac{\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}\Gamma \left(\alpha ,\theta x\right)+{\left(\theta x\right)}^{\alpha }\left(\theta x+2{\theta }^2+\alpha +1\right)e^{-\theta x}}{{\theta }^{\alpha +2}{x}^{\alpha -1}{\left(\theta +x\right)}^2{e}^{-\theta x}}$ .

It is also related to the hazard rate function as

$m\left(x\right)=\frac{1}{h\left(x\right)}$  

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 $m\left(x\right)=E\left(X-x|X>x\right)$  of WAD can be obtained as

$m\left(x;\theta ,\alpha \right)=\frac{1}{S\left(x;\theta ,\alpha \right)}{\displaystyle \underset{x}{\overset{\infty }{\int }}t{f}_1\left(t;\theta ,\alpha \right)}dt-x$  

$=\frac{1}{S\left(x;\theta ,\alpha \right)}\left[\frac{{\theta }^{\alpha +2}}{\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}\Gamma \left(\alpha \right)}{\displaystyle \underset{x}{\overset{\infty }{\int }}{t}^{\alpha }}\left(\theta +2\theta t+t^2\right)e^{-\theta t}dt\right]-x$  

                                  $=\frac{\begin{array}{l}{\left(\theta x\right)}^{\alpha }\left[\theta x+{\theta }^4+2\left(\alpha +1\right){\theta }^2+\left(\alpha +1\right)\left(\alpha +2\right)\right]e^{-\theta x}\\ +\left[\alpha {\theta }^4+\alpha \left(\alpha +1\right)\left(2{\theta }^2+\alpha +2\right)-\theta x\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}\right]\Gamma \left(\alpha ,\theta x\right)\end{array}}{\theta \left[{\left(\theta x\right)}^{\alpha }\left(\theta x+2{\theta }^2+\alpha +1\right)e^{-\theta x}+\left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}\Gamma \left(\alpha ,\theta x\right)\right]}$

It can be easily shown that $m\left(0;\theta ,\alpha \right)=\frac{\alpha \left\{{\theta }^4+2\left(\alpha +1\right){\theta }^2+\left(\alpha +1\right)\left(\alpha +2\right)\right\}}{\theta \left\{{\theta }^4+2{\theta }^2\alpha +\alpha \left(\alpha +1\right)\right\}}={\mu }_1{}^{\prime }$ .

Graphs of $m\left(x\right)$ of WAD for varying values of parameters $\theta \text{and}\alpha$  are shown in Figure 8.

Figure 8 $m\left(x\right)$ of WAD for varying values of parameters $\theta \text{and}\alpha$ .

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 $\left(X{\le }_{st}Y\right)$ if $F_X\left(x\right)\ge F_Y\left(x\right)$ for all $x$
2. hazard rate order $\left(X{\le }_{hr}Y\right)$ if $h_X\left(x\right)\ge h_Y\left(x\right)$ for all $x$
3. mean residual life order $\left(X{\le }_{mrl}Y\right)$ if $m_X\left(x\right)\le m_Y\left(x\right)$ for all $x$
4. Likelihood ratio order $\left(X{\le }_{lr}Y\right)$ if $\frac{f_X\left(x\right)}{f_Y\left(x\right)}$ decreases in $x$ .

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

$X{\le }_{lr}Y\Rightarrow X{\le }_{hr}Y\Rightarrow X{\le }_{mrl}Y$   

$\underset{X{\le }_{st}Y}{\Downarrow }$  

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 $X\sim$ WAD $\left({\theta }_1,{\alpha }_1\right)$  and $Y\sim$  WAD $\left({\theta }_2,{\alpha }_2\right)$ . If ${\alpha }_1\le {\alpha }_2\text{and}\text{}\text{}\text{}{\theta }_1>{\theta }_2$  (or ${\alpha }_1<{\alpha }_2$ ; ${\theta }_1\ge {\theta }_2$  ), then $X{\le }_{lr}Y$ and hence $X{\le }_{hr}Y$ ,$X{\le }_{mrl}Y$ and $X{\le }_{st}Y$ .

Proof: We have

$\frac{f_X\left(x;{\theta }_1,{\alpha }_1\right)}{f_Y\left(x;{\theta }_2,{\alpha }_2\right)}=\left[\frac{{\theta }_1{}^{{\alpha }_1+2}\left\{{\theta }_2{}^4+2{\theta }_2{}^2{\alpha }_2+{\alpha }_2\left({\alpha }_2+1\right)\right\}\Gamma \left({\alpha }_2\right)}{{\theta }_2{}^{{\alpha }_2+2}\left\{{\theta }_1{}^4+2{\theta }_1{}^2{\alpha }_1+{\alpha }_1\left({\alpha }_1+1\right)\right\}\Gamma \left({\alpha }_1\right)}\right]{\left(\frac{{\theta }_1+x}{{\theta }_2+x}\right)}^2{x}^{{\alpha }_1-{\alpha }_2}{e}^{-\left({\theta }_1-{\theta }_2\right)x};x>0$   

Now, taking logarithm both sides, we get

$\log \frac{f_X\left(x;{\theta }_1,{\alpha }_1\right)}{f_Y\left(x;{\theta }_2,{\alpha }_2\right)}=\log \left[\frac{{\theta }_1{}^{{\alpha }_1+2}\left\{{\theta }_2{}^4+2{\theta }_2{}^2{\alpha }_2+{\alpha }_2\left({\alpha }_2+1\right)\right\}\Gamma \left({\alpha }_2\right)}{{\theta }_2{}^{{\alpha }_2+2}\left\{{\theta }_1{}^4+2{\theta }_1{}^2{\alpha }_1+{\alpha }_1\left({\alpha }_1+1\right)\right\}\Gamma \left({\alpha }_1\right)}\right]+2\log \left(\frac{{\theta }_1+x}{{\theta }_2+x}\right)+\left({\alpha }_1-{\alpha }_2\right)\log x-\left({\theta }_1-{\theta }_2\right)x$  

 This gives $\frac{d}{dx}\ln \frac{f_X\left(x;{\theta }_1,{\alpha }_1\right)}{f_Y\left(x;{\theta }_2,{\alpha }_2\right)}=\frac{2\left({\theta }_2-{\theta }_1\right)}{\left({\theta }_1+x\right)\left({\theta }_2+x\right)}+\frac{{\alpha }_1-{\alpha }_2}{x}-\left({\theta }_1-{\theta }_2\right)$  .

Thus for ${\alpha }_1\le {\alpha }_2\text{and}\text{}\text{}\text{}{\theta }_1>{\theta }_2$  (or ${\alpha }_1<{\alpha }_2$  and ${\theta }_1\ge {\theta }_2$  ), $\frac{d}{dx}\ln \frac{f_X\left(x;{\theta }_1,{\alpha }_1\right)}{f_Y\left(x;{\theta }_2,{\alpha }_2\right)}<0$ . This means that $X{\le }_{lr}Y$ and hence $X{\le }_{hr}Y$ , $X{\le }_{mrl}Y$ and $X{\le }_{st}Y$ .

Suppose $\left(x_1,x_2,x_3,\mathrm{...},x_n\right)$  be a random sample of size $n$  from WAD. The log-likelihood function,$\log L$ of WAD can be obtained as

$\log L={\displaystyle \sum _{i=1}^n\log f_2\left(x_i;\theta ,\alpha \right)}$  

$=n\left[\left(\alpha +2\right)\log \theta -\log \left({\theta }^4+2{\theta }^2\alpha +{\alpha }^2+\alpha \right)-\log \Gamma \left(\alpha \right)\right]+\left(\alpha -1\right){\displaystyle \sum _{i=1}^n\log \left(x_i\right)}+2{\displaystyle \sum _{i=1}^n\log \left(\theta +x_i\right)}-n\theta \overline{x}$  

The maximum likelihood estimates (MLE’s) $\left(\widehat{\theta },\widehat{\alpha }\right)$  of the parameters $\left(\theta ,\alpha \right)$  of WAD are the solutions of the following log likelihood equations

$\frac{\partial \log L}{\partial \theta }=\frac{n\left(\alpha +2\right)}{\theta }-\frac{4n\left({\theta }^3+\theta \alpha \right)}{{\theta }^4+2{\theta }^2\alpha +{\alpha }^2+\alpha }+2{\displaystyle \sum _{i=1}^n\frac{1}{\theta +x_i}}-n\overline{x}=0$   

$\frac{\partial \log L}{\partial \alpha }=n\ln \theta -\frac{n\left(2{\theta }^2+2\alpha +1\right)}{{\theta }^4+2{\theta }^2\alpha +{\alpha }^2+\alpha }-n\psi \left(\alpha \right)+{\displaystyle \sum _{i=1}^n\log \left(x_i\right)}=0$   

 where $\psi \left(\alpha \right)=\frac{d}{d\alpha }\log \Gamma \left(\alpha \right)$ 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.

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 $\theta =0.5,1.0,1.5,2.0$  and $\alpha =0.5$  & $1.5$ . 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 $\theta$ and $\alpha$  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.

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 $-2\ln L$ , 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: $AIC=-2\ln L+2k$  and $\text{K-S}=\underset{x}{\text{Sup}}\left|F_n\left(x\right)-F_0\left(x\right)\right|$ , where $k$  = the number of parameters, $n$  = the sample size , $F_n\left(x\right)$ is the empirical (sample) cumulative distribution function, and $F_0\left(x\right)$  is the theoretical cumulative distribution function. The best distribution is the distribution corresponding to lower values of $-2\ln L$ , AIC, and K-S statistics.