Exponential-logarithmic distribution


In probability theory and statistics, the Exponential-Logarithmic distribution is a family of lifetime distributions with
decreasing failure rate, defined on the interval parameterized by two parameters and.

Introduction

The study of lengths of the lives of organisms, devices, materials, etc., is of major importance in the [biological">Parametric family">parameterized by two parameters and.

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 when its behavior over time is characterized by 'work-hardening' or 'immunity'.
The exponential-logarithmic model, together with its various properties, are studied by Tahmasbi and Rezaei.
This model is obtained under the concept of population heterogeneity.

Properties of the distribution

Distribution

The probability density function of the EL distribution is given by Tahmasbi and Rezaei
where and. This function is strictly decreasing in and tends to zero as. The EL distribution has its modal value of the density at x=0, given by
The EL reduces to the exponential distribution with rate parameter, as.
The cumulative distribution function is given by
and hence, the median is given by

Moments

The moment [generating function] of can be determined from the pdf by direct integration and is given by
where is a hypergeometric function. This function is also known as Barnes's extended hypergeometric function. The definition of is
where and.
The moments of can be derived from. For
, the raw moments are given by
where is the polylogarithm function which is defined as
follows:
Hence the mean and variance of the EL distribution
are given, respectively, by

The survival, hazard and mean residual life functions

The survival function and hazard function of the EL distribution are given, respectively, by
The mean residual lifetime of the EL distribution is given by
where 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:

Estimation of the parameters

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

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, then X has the exponential-logarithmic distribution in the parameterisation used above.