Exponential-logarithmic distribution

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Exponential-Logarithmic distribution (EL)
Probability density function
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[edit]

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[edit]

Distribution[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

  1. ^ a b c 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
  2. ^ Lewin, L. (1981) Polylogarithms and Associated Functions, North Holland, Amsterdam.
  3. ^ Ciumara1,Roxana; Preda2, 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