Exponential-logarithmic distribution

Exponential-Logarithmic distribution (EL)
Probability density function
Parameters	$p\in (0,1)$ $\beta >0$
Support	$x\in[0,\infty)$
PDF	${\frac {1}{-\ln p}}\times {\frac {\beta (1-p)e^{-\beta x}}{1-(1-p)e^{-\beta x}}}$
CDF	$1-{\frac {\ln(1-(1-p)e^{-\beta x})}{\ln p}}$
Mean	$-{\frac {{\text{polylog}}(2,1-p)}{\beta \ln p}}$
Median	${\frac {\ln(1+{\sqrt {p}})}{\beta }}$
Mode	0
Variance	$-{\frac {2{\text{polylog}}(3,1-p)}{\beta ^{2}\ln p}}$ $-{\frac {{\text{polylog}}^{2}(2,1-p)}{\beta ^{2}\ln ^{2}p}}$
MGF	$-{\frac {\beta (1-p)}{\ln p(\beta -t)}}{\text{hypergeom}}_{2,1}$ $([1,{\frac {\beta -t}{\beta }}],[{\frac {2\beta -t}{\beta }}],1-p)$

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 $\beta >0$ .

Introduction

The study of lengths 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)^[1] 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)^[1]

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 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 $\beta$ , 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:^[2]

\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

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

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).^[1] 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}}}.

Related distributions

The EL distribution has been generalized to form the Weibull-logarithmic distribution.^[3]

If X is defined to be the random variable which is the minimum of N independent realisations from an exponential distribution with rate paramerter 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.

References

1 2 3 Tahmasbi, R., Rezaei, S., (2008), "A two-parameter lifetime distribution with decreasing failure rate", Computational Statistics and Data Analysis, 52 (8), 3889-3901. doi:10.1016/j.csda.2007.12.002
↑ Lewin, L. (1981) Polylogarithms and Associated Functions, North Holland, Amsterdam.
↑ Ciumara, Roxana; Preda, Vasile (2009) "The Weibull-logarithmic distribution in lifetime analysis and its properties". In: L. Sakalauskas, C. Skiadas and E. K. Zavadskas (Eds.) Applied Stochastic Models and Data Analysis, The XIII International Conference, Selected papers. Vilnius, 2009 ISBN 978-9955-28-463-5

Probability distributions
List
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

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