Exponential-logarithmic distribution: Difference between revisions

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{{Short description|Family of lifetime distributions with decreasing failure rate}}

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

{{Infobox probability distribution

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

| name = <CAPTION>Exponential-Logarithmic distribution (EL)</CAPTION>

| type = continuous

<TABLE class="infobox bordered wikitable"

| pdf_image = <TH>[[File:Pdf EL.png|300px|Probability density function (pdf)</TH>]]

style="FONT-SIZE: 95%; MARGIN-BOTTOM: 0.5em; MARGIN-LEFT: 1em; WIDTH: 325px">

| cdf_image =

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

| notation =

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

| parameters = <math>p\in (0,1)</math><br><math>\beta >0</math>

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

| support <TD> = <math>x\in([0,\infty)</math></TD></TR>

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

| pdf <TD> = <math>\frac{1}{-\ln p} \times \frac{\ln(1-beta(1-p) e^{-\beta x})}{1-(1-p) e^{-\lnbeta px}}</math></TD></TR>

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

| cdf <TD> = <math>1-\frac{\text{polylog}ln(1-(2,1-p)} e^{-\beta x})}{\ln p}</math></TD></TR>

<TR vAlign=top>

| mean <TD> = <math>-\frac{2 \text{polylog}(3,1-p)}{\beta^2\ln p}-\frac{ \text{polylog}^2(2,1-p)}{\beta^2\ln^2 p}</math></TD></TR>

<TH>Parameters</TH>

| median <TD><SPAN> = <math>p\in frac{\ln(0,1+\sqrt{p})</math></SPAN><BR><SPAN> <math>}{\beta >0}</math></SPAN></TD></TR>

| mode = 0

<TR>

| variance = <math>-\frac{2 \text{polylog}(3,1-p)}{\beta^2\ln p}</math><br> <math>-\frac{ \text{polylog}^2(2,1-p)}{\beta^2\ln^2 p}</math>

<TH>Support</TH>

| skewness =

<TD><math>x\in(0,\infty)</math></TD></TR>

| kurtosis =

<TR>

| entropy =

<TH>Probability density function (pdf)</TH>

| mgf <TD> = <math>-\frac{\beta(1-p)}{-\ln p (\beta-t)} \timestext{hypergeom}_{2,1} </math><br> <math>([1,\frac{\beta(1-p) e^t}{\beta}],[\frac{2\beta-t}{\beta}],1-p)</math>

| cf =

x}}{1-(1-p) e^{-\beta x}}</math></TD></TR>

| pgf =

<TR>

| fisher =

<TH>Cumulative distribution function (cdf)</TH>

}}

<TD><math>1-\frac{\ln(1-(1-p) e^{-\beta x})}{\ln p}</math></TD></TR>

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

<TR>

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

<TH>Mean</TH>

<TD><math>-\frac{\text{polylog}(2,1-p)}{\beta\ln p}</math></TD></TR>

<TR>

<TH>Median</TH>

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

<TR>

<TH>Mode</TH>

<TD>0</TD></TR>

<TR>

<TH>Variance</TH>

<TD><math>-\frac{2 \text{polylog}(3,1-p)}{\beta^2\ln p}-\frac{ \text{polylog}^2(2,1-p)}{\beta^2\ln^2 p}</math></TD></TR>

<TR>

<TH>Skewness</TH>

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

<TR>

<TH>Excess kurtosis</TH>

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

<TR>

<TH>Moment-generating function (mgf)</TH>

<TD><math>-\frac{\beta(1-p)}{\ln p (\beta-t)}</math><br> <math> \text{hypergeom}_{2,1}([1,\frac{\beta-t}{\beta}],[\frac{2\beta-t}{\beta}],1-p)</math></TD></TR>

<TR>

<TH>Characteristic function</TH>

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

</TABLE>

== 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).<ref name="tahmasbi2008">Tahmasbi, R., Rezaei, S., (2008), "A two-parameter lifetime distribution with decreasing failure rate", ''Computational Statistics and Data Analysis'', Vol. 52 (8), pp. 3889-3901. {{doi|10.1016/j.csda.2007.12.002}}</ref>

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

compounding).

== Properties of the distribution ==

=== Distribution ===

The [[probability density function]] (pdf) of the EL distribution is given by{{cn}] Tahmasbi and Rezaei (2008)<ref name="tahmasbi2008"/>

:<math> f(x; p, \beta) := \left( \frac{1}{-\ln p}\right) \frac{\beta(1-p)e^{-\beta x}}{1-(1-p)e^{-\beta x}} </math>

where <math>p\in (0,1)</math> and <math>\beta >0</math>. This function is strictly decreasing in <math>x</math> and tends to zero as <math>x\rightarrow \infty</math>. The EL distribution has its [[modeMode (statistics)|modal value]] given,of the density at x=0, given by

:<math>\frac{\beta (1-p)}{-p \ln p}</math>

The EL reduces to the [[exponential distribution]] with rate parameter <math>\beta</math>, as <math>p\rightarrow 1</math>.

The [[cumulative distribution function]] is given by

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: <math>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)}</math>

where <math>n=[n_1, n_2,\dots , n_N]</math> and <math>{d}=[d_1, d_2, \dots , d_D]</math>.

The moments of <math>X</math> can be derived from <math>M_X(t)</math>. For

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:<math>E(X^r;p,\beta)=-r!\frac{\operatorname{Li}_{r+1}(1-p) }{\beta^r\ln p},</math>

where <math>\operatorname{Li}_a(z)</math> is the [[polylogarithm]] function which is defined as

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

Holland, Amsterdam.</ref>

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

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

[[File:Hazard EL.png|thumb|300px|Hazard function]]

The [[survival function]] (also known as the reliability

function) and [[hazard function]] (also known as the failure rate

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== Estimation of the parameters ==

To estimate the parameters, the [[Expectation-maximization algorithm|EM algorithm]] is used. This method is discussed by Tahmasbi and Rezaei (2008).<ref name="tahmasbi2008"/> 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>

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: <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>

==Related distributions==

The EL distribution has been generalized to form the Weibull-logarithmic distribution.<ref>Ciumara, Roxana; Preda, Vasile (2009) [https://www.proquest.com/openview/7f1efa684243ce36231867620f09373a/1 "The Weibull-logarithmic distribution in lifetime analysis and its properties"]. In: L. Sakalauskas, C. Skiadas and

E. K. Zavadskas (Eds.) [http://www.vgtu.lt/leidiniai/leidykla/ASMDA_2009/ ''Applied Stochastic Models and Data Analysis''] {{Webarchive|url=https://web.archive.org/web/20110518043330/http://www.vgtu.lt/leidiniai/leidykla/ASMDA_2009/ |date=2011-05-18 }}, The XIII International Conference, Selected papers. Vilnius, 2009 {{ISBN|978-9955-28-463-5}}</ref>

If ''X'' is defined to be the [[random variable]] which is the minimum of ''N'' independent realisations from an [[exponential distribution]] with rate parameter ''&beta;'', and if ''N'' is a realisation from a [[logarithmic distribution]] (where the parameter ''p'' in the usual parameterisation is replaced by {{nowrap|1=(1&nbsp;&minus;&nbsp;''p'')}}), then ''X'' has the exponential-logarithmic distribution in the parameterisation used above.

==References==

{{Reflist}}

{{ProbDistributions|continuous-semi-infinite}}

[[Category:Continuous distributions]]