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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in(0,1)} and $\beta >0$.

Exponential-Logarithmic distribution (EL)

Probability density function
Parameters	$p\in (0,1)$ $\beta >0$
Support	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in[0,\infty)}
PDF	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{-\ln p} \times \frac{\beta(1-p) e^{-\beta x}}{1-(1-p) e^{-\beta x}}}
CDF	$1-{\frac {\ln(1-(1-p)e^{-\beta x})}{\ln p}}$
Mean	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\text{polylog}(2,1-p)}{\beta\ln p}}
Median	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\ln(1+\sqrt{p})}{\beta}}
Mode	0
Variance	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{2 \text{polylog}(3,1-p)}{\beta^2\ln p}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{ \text{polylog}^2(2,1-p)}{\beta^2\ln^2 p}}
MGF	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\beta(1-p)}{\ln p (\beta-t)} \text{hypergeom}_{2,1} } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ([1,\frac{\beta-t}{\beta}],[\frac{2\beta-t}{\beta}],1-p)}

Introduction

The study of lengths of the lives of organisms, devices, materials, etc., is of major importance in the biological and engineering sciences. In general, the lifetime 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 Tahmasbi and Rezaei (2008)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x; p, \beta) := \left( \frac{1}{-\ln p}\right) \frac{\beta(1-p)e^{-\beta x}}{1-(1-p)e^{-\beta x}} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in (0,1)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta >0} . This function is strictly decreasing in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and tends to zero as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\rightarrow \infty} . The EL distribution has its modal value of the density at x=0, given by

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

The EL reduces to the exponential distribution with rate parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} , as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\rightarrow 1} .

The cumulative distribution function is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x;p,\beta)=1-\frac{\ln(1-(1-p) e^{-\beta x})}{\ln p},}

and hence, the median is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_\text{median}=\frac{\ln(1+\sqrt{p})}{\beta}} .

Moments

The moment generating function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} can be determined from the pdf by direct integration and is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{2,1} } is a hypergeometric function. This function is also known as Barnes's extended hypergeometric function. The definition of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=[n_1, n_2,\dots , n_N]} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {d}=[d_1, d_2, \dots , d_D]} .

The moments of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} can be derived from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_X(t)} . For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\in\mathbb{N}} , the raw moments are given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(X^r;p,\beta)=-r!\frac{\operatorname{Li}_{r+1}(1-p) }{\beta^r\ln p},}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Li}_a(z)} is the polylogarithm function which is defined as follows:

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

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(X)=-\frac{\operatorname{Li}_2(1-p)}{\beta\ln p},}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Hazard function

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(x)=\frac{\ln(1-(1-p)e^{-\beta x})}{\ln p},}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta^{(h+1)} = n \left( \sum_{i=1}^n\frac{x_i}{1-(1-p^{(h)})e^{-\beta^{(h)}x_i}} \right)^{-1},}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}}.}

Related distributions

The EL distribution has been generalized to form the Weibull-logarithmic distribution.

If X is defined to be the random variable which is the minimum of N independent realisations from an exponential distribution with rate parameter β, and if N is a realisation from a logarithmic distribution (where the parameter p in the usual parameterisation is replaced by (1 − p)), then X has the exponential-logarithmic distribution in the parameterisation used above.