Exponential-logarithmic distribution

In probability theory and statistics, the exponential-logarithmic (EL) distribution is a family of lifetime distributions with decreasing failure rate, defined on the interval (0, ∞). This distribution is parameterized by two parameters $p\in (0,1)$ and $\beta >0$.

Exponential-Logarithmic distribution (EL)

Probability density function
Hazard function
Parameters	$p\in (0,1)$ $\beta >0$
Support	$x\in (0,\infty )$
Probability density function (pdf)	${\frac {1}{-\ln p}}\times {\frac {\beta (1-p)e^{-\beta x}}{1-(1-p)e^{-\beta x}}}$
Cumulative distribution function (cdf)	$1-{\frac {\ln(1-(1-p)e^{-\beta x})}{\ln p}}$
Mean	$-{\frac {{\text{polylog}}(2,1-p)}{\beta \ln p}}$
Median	${\frac {\ln(1+{\sqrt {p}})}{\beta }}$
Mode	0
Variance	$-{\frac {2{\text{polylog}}(3,1-p)}{\beta ^{2}\ln p}}-{\frac {{\text{polylog}}^{2}(2,1-p)}{\beta ^{2}\ln ^{2}p}}$
Skewness
Excess kurtosis
Moment-generating function (mgf)	$-{\frac {\beta (1-p)}{\ln p(\beta -t)}}$ ${\text{hypergeom}}_{2,1}([1,{\frac {\beta -t}{\beta }}],[{\frac {2\beta -t}{\beta }}],1-p)$
Characteristic function

Introduction

The study of lengths of organisms, devices, materials, etc., is of major importance in the biological and engineering sciences. In general, the life time of a device is expected to exhibit decreasing failure rate (DFR) when its behavior over time is characterized by 'work-hardening' (in engineering terms) or 'immunity' (in biological terms).

The exponential-logarithmic model, together with its various properties, are studied by Tahmasbi and Rezaei (2008) This model is obtained under the concept of population heterogeneity (through the process of compounding).

Properties of the distribution

Distribution

The probability density function (pdf) of the EL distribution is given by

$f(x;p,\beta ):=\left({\frac {1}{-\ln p}}\right){\frac {\beta (1-p)e^{\beta x}}{1-(1-p)e^{\beta x}}}$

where $p\in (0,1)$ and $\beta >0$. This function is strictly decreasing in $x$ and tends to zero as $x\rightarrow \infty$. The EL distribution has modal value given, at x=0, by

${\frac {\beta (1-p)}{-p\ln p}}$

The EL reduces to the exponential distribution with parameter $\beta$, as $p\rightarrow 1$.

The cumulative distribution function is given by

$F(x;p,\beta )=1-{\frac {\ln(1-(1-p)e^{-\beta x})}{\ln p}},$

and hence, the median is given by

$x_{\text{median}}={\frac {\ln(1+{\sqrt {p}})}{\beta }}$.

Moments

The moment generating function of $X$ can be determined from the pdf by direct integration and is given by

$M_{X}(t)=E(e^{tX})=-{\frac {\beta (1-p)}{\ln p(\beta -t)}}F_{2,1}\left(\left[1,{\frac {\beta -t}{\beta }}\right],\left[{\frac {2\beta -t}{\beta }}\right],1-p\right),$

where $F_{2,1}$ is a hypergeometric function. This function is also known as Barnes's extended hypergeometric function. The definition of $F_{N,D}({n,d},z)$ is

$F_{N,D}(n,d,z):=\sum _{k=0}^{\infty }{\frac {z^{k}\prod _{i=1}^{p}\Gamma (n_{i}+k)\Gamma ^{-1}(n_{i})}{\Gamma (k+1)\prod _{i=1}^{q}\Gamma (d_{i}+k)\Gamma ^{-1}(d_{i})}}$

where $n=[n_{1},n_{2},\dots ,n_{N}]$ and ${d}=[d_{1},d_{2},\dots ,d_{D}]$.

The moments of $X$ can be derived from $M_{X}(t)$. For $r\in \mathbb {N}$, the raw moments are given by

$E(X^{r};p,\beta )=-r!{\frac {\operatorname {Li} _{r+1}(1-p)}{\beta ^{r}\ln p}},$

where $\operatorname {Li} _{a}(z)$ is the polylogarithm function which is defined as follows (Lewin, 1981)

$\operatorname {Li} _{a}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{a}}}.$

Hence the mean and variance of the EL distribution are given, respectively, by

$E(X)=-{\frac {\operatorname {Li} _{2}(1-p)}{\beta \ln p}},$

$\operatorname {Var} (X)=-{\frac {2\operatorname {Li} _{3}(1-p)}{\beta ^{2}\ln p}}-\left({\frac {\operatorname {Li} _{2}(1-p)}{\beta \ln p}}\right)^{2}.$

The survival, hazard and mean residual life functions

The survival function (also known as the reliability function) and hazard function (also known as the failure rate function) of the EL distribution are given, respectively, by

$s(x)={\frac {\ln(1-(1-p)e^{-\beta x})}{\ln p}},$

$h(x)={\frac {-\beta (1-p)e^{-\beta x}}{(1-(1-p)e^{-\beta x})\ln(1-(1-p)e^{-\beta x})}}.$

The mean residual lifetime of the EL distribution is given by

$m(x_{0};p,\beta )=E(X-x_{0}|X\geq x_{0};\beta ,p)=-{\frac {\operatorname {Li} _{2}(1-(1-p)e^{-\beta x_{0}})}{\beta \ln(1-(1-p)e^{-\beta x_{0}})}}$

where $\operatorname {Li} _{2}$ is the dilogarithm function

Random number generation

Let U be a random variate from the standard uniform distribution. Then the following transformation of U has the EL distribution with parameters p and β:

$X={\frac {1}{\beta }}\ln \left({\frac {1-p}{1-p^{U}}}\right).$

Estimation of the parameters

To estimate the parameters, the EM algorithm is used. This method is discussed by Tahmasbi and Rezaei (2008). The EM iteration is given by

$\beta ^{(h+1)}=n\left(\sum _{i=1}^{n}{\frac {x_{i}}{1-(1-p^{(h)})e^{-\beta ^{(h)}x_{i}}}}\right)^{-1},$

$p^{(h+1)}={\frac {-n(1-p^{(h+1)})}{\ln(p^{(h+1)})\sum _{i=1}^{n}\{1-(1-p^{(h)})e^{-\beta ^{(h)}x_{i}}\}^{-1}}}.$