In this paper some of the important mathematical properties including moment generating function, mean deviations, order statistics, Bonferroni and Lorenz curves, Renyi entropy and stress strength reliability of two-parameter Lindley distribution (TPLD) of Shanker & Mishra¹ have been discussed. Its goodness of fit over exponential and Lindley distributions have been illustrated with some real lifetime data-sets and found that TPLD is preferable over exponential and Lindley distributions for modeling lifetime data-sets.

Keywords: mean deviations; order statistics, bonferroni and lorenz curves, entropy, stress-strength reliability, goodness of fit

The probability density function (p.d.f.) and the cumulative distribution function (c.d.f.) of distribution, introduced in the context of Bayesian analysis as a counter example of fiducial statistics, are given by

The detailed study about its mathematical properties, estimation of parameter and application showing the superiority of Lindley distribution over exponential distribution for the waiting times before service of the bank customers has been done by Ghitany et al.² The Lindley distribution has been generalized extended and modified by different researchers including^1,3-19 are some among others.

The probability density function (p.d.f.) and cumulative distribution function (c.d.f) of two-parameter Lindley distribution (TPLD) of Shanker & Mishra¹ are given by

At $\alpha =1$ , both (1.3) and (1.4) reduce to the corresponding expressions (1.1) and (1.2) of Lindley distribution. The first two moments about origin and the variance of TPLD of Shanker & Mishra¹ are given by

At $\alpha =1$ , these moments reduce to the corresponding moments of Lindley distribution. Shanker & Mishra¹ have derived and discussed some of its mathematical properties such as shape, moments, coefficient of variation, coefficient of skewness and kurtosis, hazard rate function, mean residual life function and stochastic orderings. They have also discussed the estimation of its parameters using maximum likelihood estimation and method of moments and its goodness of fit over Lindley distribution. It has been observed that many important mathematical properties of this distribution have not been studied.

In the present paper some of the important mathematical properties including moment generating function, mean deviations, order statistics, Bonferroni and Lorenz curves, Renyi entropy and stress strength reliability of TPLD of Shanker & Mishra¹ have been derived and discussed. Its goodness of fit over exponential and Lindley distributions have been illustrated with some real lifetime data-sets and found that TPLD gives better fit than exponential and Lindley distributions.

The moment generating function, $\left(M_X\left(t\right)\right)$  of TPLD (1.3) can be obtained as

 It can be easily seen that the expression for ${\mu }_r{}^{\prime }$ obtained as the coefficient of $\frac{t^r}{r!}$  is given as

For $\alpha =1$ , ${\mu }_r{}^{\prime }$ reduces to the corresponding ${\mu }_r{}^{\prime }$ of Lindley distribution.

The amount of scatter in a population is measured to some extent by the totality of deviations usually from mean and median. These are known as the mean deviation about the mean and the mean deviation about the median defined by ${\delta }_1\left(X\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}\left|x-\mu \right|}f\left(x\right)dx$  and ${\delta }_2\left(X\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}\left|x-M\right|}f\left(x\right)dx$ , respectively, where $\mu =E\left(X\right)$  and $$M=\text{Median }\left(X\right)$$ . The measures ${\delta }_1\left(X\right)$  and ${\delta }_2\left(X\right)$ can be calculated using the relationships

and

Using p.d.f. (1.3), and expression for mean of two-parameter Lindley distribution, we have

${\displaystyle \underset{0}{\overset{\mu }{\int }}xf\left(x\right)}dx=\mu -\frac{\left\{{\theta }^2\left({\mu }^2+\alpha \mu \right)+2\theta \mu +\left(\alpha \theta +2\right)\right\}{e}^{-\theta \mu }}{\theta \left(\alpha \theta +1\right)}$     (3.3)

Using expressions from (3.1), (3.2) and (3.3), and little algebraic simplification, the mean deviation about mean, ${\delta }_1\left(X\right)$ and the mean deviation about median, ${\delta }_2\left(X\right)$ of TPLD (1.3) are obtained as

${\delta }_1\left(X\right)=\frac{2\left(\theta \mu +\alpha \theta +2\right)e^{-\theta \mu }}{\theta \left(\alpha \theta +1\right)}$     (3.4)

and ${\delta }_2\left(X\right)=\frac{2\left\{{\theta }^2\left(M^2+\alpha M\right)+2\theta M+\left(\alpha \theta +2\right)\right\}{e}^{-\theta M}}{\theta \left(\alpha \theta +1\right)}-\mu$     (3.5)

It can be easily seen that expressions (3.4) and (3.5) of TPLD (1.3) reduce to the corresponding expressions of Lindley distribution at $\alpha =1$ .

Let $X_1,X_2,\mathrm{...},X_n$  be a random sample of size  from two-parameter Lindley distribution (1.3). Let $X_{\left(1\right)}<X_{\left(2\right)}<\mathrm{...}<X_{\left(n\right)}$ denote the corresponding order statistics. The p.d.f. and the c.d.f. of the $k$ th order statistic, say $Y=X_{\left(k\right)}$ are given by

and

$F_Y\left(y\right)={\displaystyle \sum _{j=k}^n\left(\begin{array}{c}n\\ j\end{array}\right)}{F}^j\left(y\right){\left\{1-F\left(y\right)\right\}}^{n-j}$

$={\displaystyle \sum _{j=k}^n{\displaystyle \sum _{l=0}^{n-j}\left(\begin{array}{c}n\\ j\end{array}\right)}\left(\begin{array}{c}n-j\\ l\end{array}\right)}{\left(-1\right)}^l{F}^{j+l}\left(y\right)$

respectively, for $k=1,2,3,\mathrm{...},n$

Thus, the p.d.f. and the c.d.f of the th order statistics of TPLD (1.3) are obtained as

$f_Y\left(y\right)=\frac{n!{\theta }^2\left(\alpha +x\right)e^{-\theta x}}{\left(\alpha \theta +1\right)\left(k-1\right)!\left(n-k\right)!}{\displaystyle \sum _{l=0}^{n-k}\left(\begin{array}{c}n-k\\ l\end{array}\right)}{\left(-1\right)}^l\times {\left[1-\frac{1+\alpha \theta +\theta \text{}x}{\alpha \theta +1}{e}^{-\theta x}\right]}^{k+l-1}$

and

$F_Y\left(y\right)={\displaystyle \sum _{j=k}^n{\displaystyle \sum _{l=0}^{n-j}\left(\begin{array}{c}n\\ j\end{array}\right)}\left(\begin{array}{c}n-j\\ l\end{array}\right)}{\left(-1\right)}^l{\left[1-\frac{1+\alpha \theta +\theta \text{}x}{\alpha \theta +1}{e}^{-\theta x}\right]}^{j+l}$

It can be easily verified that the expressions for the p.d.f. and c.d.f. of the th order statistics of TPLD (1.3) reduce to the expressions for the p.d.f. and c.d.f. of the th order statistics of Lindley distribution at $\alpha =1$

The Bonferroni and Lorenz curves²⁰ and Bonferroni and Gini indices have applications not only in economics to study income and poverty, but also in other fields like reliability, demography, insurance and medicine. The Bonferroni and Lorenz curves are defined as

$B\left(p\right)=\frac{1}{p\mu }{\displaystyle \underset{0}{\overset{q}{\int }}xf\left(x\right)}dx=\frac{1}{p\mu }\left[{\displaystyle \underset{0}{\overset{\infty }{\int }}xf\left(x\right)dx-}{\displaystyle \underset{q}{\overset{\infty }{\int }}xf\left(x\right)}dx\right]=\frac{1}{p\mu }\left[\mu -{\displaystyle \underset{q}{\overset{\infty }{\int }}xf\left(x\right)}dx\right]$ (5.1)

$L\left(p\right)=\frac{1}{\mu }{\displaystyle \underset{0}{\overset{q}{\int }}xf\left(x\right)}dx=\frac{1}{\mu }\left[{\displaystyle \underset{0}{\overset{\infty }{\int }}xf\left(x\right)dx-}{\displaystyle \underset{q}{\overset{\infty }{\int }}xf\left(x\right)}dx\right]=\frac{1}{\mu }\left[\mu -{\displaystyle \underset{q}{\overset{\infty }{\int }}xf\left(x\right)}dx\right]$ (5.2)

respectively or equivalently

$B\left(p\right)=\frac{1}{p\mu }{\displaystyle \underset{0}{\overset{p}{\int }}{F}^{-1}\left(x\right)}dx$ (5.3)

and $L\left(p\right)=\frac{1}{\mu }{\displaystyle \underset{0}{\overset{p}{\int }}{F}^{-1}\left(x\right)}dx$ (5.4)

respectively, where $\mu =E\left(X\right)$ and $q=F^{-1}\left(p\right)$ .

The Bonferroni and Gini indices are thus defined as

$B=1-{\displaystyle \underset{0}{\overset{1}{\int }}B\left(p\right)}dp$ (5.5)

and $G=1-2{\displaystyle \underset{0}{\overset{1}{\int }}L\left(p\right)}dp$ (5.6)

respectively.

Using p.d.f. (1.3), we get

${\displaystyle \underset{q}{\overset{\infty }{\int }}xf\left(x\right)}dx=\frac{\left\{{\theta }^2\left(q^2+\alpha q\right)+2\theta q+\left(\alpha \theta +2\right)\right\}{e}^{-\theta q}}{\theta \left(\alpha \theta +1\right)}$ (5.7)

Now using equation (5.7) in (5.1) and (5.2), we get

$B\left(p\right)=\frac{1}{p}\left[1-\frac{\left\{{\theta }^2\left(q^2+\alpha q\right)+2\theta q+\left(\alpha \theta +2\right)\right\}{e}^{-\theta q}}{\alpha \theta +2}\right]$ (5.8)

and $L\left(p\right)=1-\frac{\left\{{\theta }^2\left(q^2+\alpha q\right)+2\theta q+\left(\alpha \theta +2\right)\right\}{e}^{-\theta q}}{\alpha \theta +2}$ (5.9)

Now using equations (5.8) and (5.9) in (5.5) and (5.6), the Bonferroni and Gini indices of TPLD (1.3) are obtained as

$B=1-\frac{\left\{{\theta }^2\left(q^2+\alpha q\right)+2\theta q+\left(\alpha \theta +2\right)\right\}{e}^{-\theta q}}{\alpha \theta +2}$ (5.10)

$G=-1+\frac{2\left\{{\theta }^2\left(q^2+\alpha q\right)+2\theta q+\left(\alpha \theta +2\right)\right\}{e}^{-\theta q}}{\alpha \theta +2}$ (5.11)

The Bonferroni and Gini indices of Lindley distribution are particular cases of the Bonferroni and Gini indices (5.10) and (5.11) of TPLD (1.3) for $\alpha =1$ .

An entropy of a random variable is a measure of variation of uncertainty. A popular entropy measure is Renyi entropy.²¹ If is a continuous random variable having probability density function $f(.)$ , then Renyi entropy is defined as

$T_R\left(\gamma \right)=\frac{1}{1-\gamma }\log \left\{{\displaystyle \int f^{\gamma }\left(x\right)dx}\right\}$

where $\gamma >0\text{and}\gamma \ne 1$ .

Thus, the Renyi entropy for TPLD (1.3) can be obtained as

The Renyi entropy of Lindley distribution is a particular case of the Renyi entropy TPLD at $\alpha =1$ .

The stress- strength reliability describes the life of a component which has random strength that is subjected to a random stress . When the stress applied to it exceeds the strength, the component fails instantly and the component will function satisfactorily till $X>Y$ . Therefore, $R=P\left(Y<X\right)$ is a measure of component reliability and in statistical literature it is known as stress-strength parameter. It has wide applications in almost all areas of knowledge especially in engineering such as structures, deterioration of rocket motors, static fatigue of ceramic components, aging of concrete pressure vessels etc.

Let $X$ and $Y$ be independent strength and stress random variables having TPLD (1.3) with parameter $\left({\alpha }_1,{\theta }_1\right)$ and $\left({\alpha }_2,{\theta }_2\right)$ respectively. Then the stress-strength reliability $R$ is obtained as

$R=P\left(Y<X\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}P\left(Y<X|X=x\right)}{f}_X\left(x\right)dx$

$={\displaystyle \underset{0}{\overset{\infty }{\int }}f\left(x;{\alpha }_1,{\theta }_1\right)}F\left(x;{\alpha }_2,{\theta }_2\right)dx$

$=1-\frac{{\theta }_1{}^2\left[2{\theta }_2+\left({\alpha }_1{\theta }_2+{\alpha }_2{\theta }_2+1\right)\left({\theta }_1+{\theta }_2\right)+{\alpha }_1\left({\alpha }_2{\theta }_2+1\right){\left({\theta }_1+{\theta }_2\right)}^2\right]}{\left({\alpha }_1{\theta }_1+1\right)\left({\alpha }_2{\theta }_2+1\right){\left({\theta }_1+{\theta }_2\right)}^2}$

The expression of stress-strength reliability of Lindley distribution is a particular case of the expression of stress-strength reliability of TPLD (1.3) at ${\alpha }_1={\alpha }_2=1$ .