Exponential-logarithmic distribution: Difference between revisions

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In probability theory and statistics, the '''Exponential-Logarithmic (EL) distribution'''

is a family of lifetime distribution with<br>

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<CAPTION>Exponential-Logarithmic distribution (EL)</CAPTION>

<TR style="TEXT-ALIGN: center">

<TD colSpan=2>Probability density function<BR>[[File:Pdf_ELPdf EL.png]]</TD></TR>

<TR style="TEXT-ALIGN: center">

<TD colSpan=2>Hazard function<BR>[[File:Hazard_ELHazard EL.png]]</TD></TR>

<TR vAlign=top>

<TH>Parameters</TH>

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<TD>&nbsp;</TD></TR>

</TABLE>

[table of contents]

== Introduction ==

<p>The study of length of organisms, devices,

materials, etc., is of major importance in the biological and

engineering science. In general, life time of an device is

expected to exhibit decreasing failure rate (DFR) when its

behavior over time is characterized by 'work-hardening' (in

engineering term) or 'immunity' (in biological term).</p>

<p>The Exponential-Logarithmic model together with its various properties are studied in

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).</p>

== Properties of the distribution ==

=== Distribution ===

<p>The probability density function of EL

distribution is monotone decreasing with

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

<p>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

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

<p>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 obtained by

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

=== Moments ===

<p>The moment generating function of <math>X</math> is determined from pdf by

direct integration and is given by <math>M_X(t)=E(e^{tX})=-\frac{\beta(1-p)}{\ln p (\beta-t)} hypergeom_{2,1}([1,\frac{\beta-t}{\beta}],[\frac{2\beta-t}{\beta}],1-p),</math><br>

where <math>hypergeom(.)</math> is hypergeometric function. This function

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function is quickly evaluated and readily available in standard

software such as Maple.

</p>

<p>

The moments of <math>X</math> are determined from derivation of <math>M_X(t)</math>. For

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

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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></p>

<p>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><br>

<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>

</p>

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

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<math>h(x)=\frac{-\beta(1-p)e^{-\beta x}}{(1-(1-p)e^{-\beta x})\ln(1-(1-p)e^{-\beta x})}.</math>

<p>

The mean residual lifetime

of the EL distribution is given by<br/>

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follows:<br/>

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

</p>

=== Random number generation ===

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parameters <math>p</math> and <math>\beta</math>:<br>

<math>X=\frac{1}{\beta}\ln(\frac{1-p}{1-p^U}).</math>

== Estimation of the parameters ==

To estimate the parametres, EM algorithm is used. This method is disscussed in Tahmasbi and Rezaei (2008). The EM iteration is given by

<br/>

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

<math>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}}.</math><br/>

==References==

{{Reflist}}

<br/>

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