Exponential-logarithmic distribution: Difference between revisions

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{{Wikify|date=August 2009}}

In [[probability theory]] and [[statistics]], the '''exponential-logarithmic (EL) distribution''' is a family of lifetime [[probability distribution|distributions]] with

decreasing [[failure rate]], defined on the interval&nbsp;(0,&nbsp;&infin;). This distribution is [[Parametric family|parameterized]] by two parameters <math>p\in(0,1)</math> and <math>\beta >0</math>.

<TABLE class="infobox bordered wikitable"

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== Introduction ==

The study of lengths of organisms, devices, materials, etc., is of major importance in the [[biological]] and [[engineering]] sciencesciences. In general, the life time of ana device is expected to exhibit decreasing failure rate (DFR) when its behavior over time is characterized by 'work-hardening' (in engineering termterms) or 'immunity' (in biological termterms).

The exponential-logarithmic model, together with its various properties, are studied inby Tahmasbi and Rezaei (2008)<ref>Tahmasbi, R., Rezaei, S., 2008, "A two-parameter lifetime distribution with decreasing failure rate", Computational Statistics and Data Analysis, Vol. 52, pp. 3889-3901.</ref>

This model is obtained under the concept of population heterogeneity (through the process of

compounding).

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=== Distribution ===

The [[probability density function]] (pdf) of the EL

distribution is monotone decreasing with

modal value <math>\beta(1-p)(-p \ln p)^{-1}</math> at <math>x=0</math>.

For all values of parameters, the pdf is strictly decreasing in

<math>x</math> and tending to zero as <math>x\rightarrow \infty</math>. The EL leads to the

exponential distribution with parameter <math>\beta</math>, as <math>p\rightarrow 1</math>.

The distribution function is given by <br>

:<math>F_X(x;p,\beta)=1-\frac{\ln(1-(1-p) e^{-\beta x})}{\ln p},</math><br>
and hence, the median is obtainedgiven by

:<math>\mathrm{median}=\frac{\ln(1+\sqrt{p})}{\beta}</math>.

=== Moments ===

The [[moment generating function]] of <math>X</math> iscan be determined from the pdf by

direct integration and is given by

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software such as Maple.

The moments of <math>X</math> arecan determinedbe derived from derivation of <math>M_X(t)</math>. For

<math>r\in\mathbb{N}</math>, the raw moments are given by<br>

:<math>E(X^r;p,\beta)=-\frac{r! polylog(r+1,1-p)}{\beta^r\ln p}, r\in\mathbb{N},</math><br>

where <math>polylog(.)</math> is the polylogarithm function and itwhich is defined as

follows (Lewin, 1981) <ref>Lewin, L., 1981, Polylogarithms and Associated Functions, North

Holland, Amsterdam.</ref>:<br>

:<math>polylog(a, z) =\sum_{k=1}^{\infty}\frac{z^k}{k^a}.</math>

Hence the [[mean]] and [[variance]] of the EL distribution

are given, respectively, by<br>

:<math>E(X)=-\frac{polylog(2,1-p)}{\beta\ln p},</math>

:<math>Var(X)=-\frac{2 polylog(3,1-p)}{\beta^2\ln p}-\frac{ polylog^2(2,1-p)}{\beta^2\ln^2 p}.</math>

=== The survival, hazard and mean residual life functions ===

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

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: <math>m(x_0;p,\beta)=E(X-x_0|X\geq x_0;\beta,p)=-\frac{\operatorname{dilog}(1-(1-p)e^{-\beta x_0})}{\beta \ln(1-(1-p)e^{-\beta x_0})}</math>

where dilog is the [[dilogarithm]] function defined as follows:

: <math>\operatorname{dilog}(a)=\int_1^a \frac{\ln(x)}{1-x} \, dx.</math>

=== Random number generation ===

Let ''U'' be a [[random variablevariate]] from the standard [[Uniform distribution (continuous)|uniform distribution]].

Then the following transformation of ''U'' has the EL distribution with

parameters ''p'' and&nbsp;''&beta;'':

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== Estimation of the parameters ==

To estimate the parameters, the [[Expectation-maximization algorithm|EM algorithm]] is used. This method is discussed inby Tahmasbi and Rezaei (2008). The EM iteration is given by

: <math>\beta^{(h+1)} = n \left( \sum_{i=1}^n\frac{x_i}{1-(1-p^{(h)})e^{-\beta^{(h)}x_i}} \right)^{-1},</math>