Exponential-logarithmic distribution
Introduction In probability theory and statistics, the '''Exponential-Logarithmic (EL)''' distribution is a family of lifetime distributions withdecreasing failure rate, defined on the interval 0, ∞). This distribution is 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 (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 ofcompounding).
Properties of the distribution === Distribution ===The probability density function (pdf) of the EL distribution is given by Tahmasbi and Rezaei (2008):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 asfollows::Hence the mean and variance of the EL distributionare given, respectively, by::=== The survival, hazard and mean residual life functions ===Hazard functionThe survival function (also known as the reliabilityfunction) and hazard function (also known as the failure ratefunction) 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 withparameters ''p'' and ''β''::
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: :
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 ), then ''X'' has the exponential-logarithmic distribution in the parameterisation used above.
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