Exponential-logarithmic distribution: Difference between revisions

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

{{Wikify|date=August 2009}}

{{Infobox probability distribution

| name = Exponential-Logarithmic distribution (EL)

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

| type = continuous

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

| pdf_image = [[File:Pdf EL.png|300px|Probability density function]]

| cdf_image =

<TABLE class="infobox bordered wikitable"

| notation =

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

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

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

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

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

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

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

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

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

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

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

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

<TR vAlign=top>

| mode = 0

<TH>Parameters</TH>

| variance <TD><SPAN>= <math>p-\infrac{2 \text{polylog}(03,1-p)}{\beta^2\ln p}</math></SPAN><BR><SPANbr> <math>-\frac{ \text{polylog}^2(2,1-p)}{\beta^2\ln^2 >0p}</math></SPAN></TD></TR>

| skewness =

<TR>

| kurtosis =

<TH>Support</TH>

| entropy =

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

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

<TR>

| cf =

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

| pgf =

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

| fisher =

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

}}

<TR>

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

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

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>.

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

<TR>

<TH>Mean</TH>

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

[table of contents]

== Introduction ==

The study of lengths of the lives of organisms, devices, materials, etc., is of major importance in the [[biological]] and [[engineering]] sciencesciences. In general, lifethe timelifetime 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 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 Tahmasbi and Rezaei (2008)<ref name="tahmasbi2008"/>

distribution is monotone decreasing with

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

:<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>

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

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

exponential distribution with parameter :<math>\frac{\beta</math>, as(1-p)}{-p <math>p\rightarrowln 1p}</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 <br>

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

:<math>x_\text{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

: <math>M_X(t) = E(e^{tX}) = -\frac{\beta(1-p)}{\ln p (\beta-t)} F_{2,1}\left(\left[1,\frac{\beta-t}{\beta}\right],\left[\frac{2\beta-t}{\beta}\right],1-p\right),</math>

direct integration and is given by

where <math>F_{2,1} </math> is a [[hypergeometric function]]. This function is also known as ''Barnes's extended hypergeometric function''. The definition of <math>F_{N,D}({n,d},z)</math> is

: <math>M_X(t) = E(e^{tX}) = -\frac{\beta(1-p)}{\ln p (\beta-t)} \operatorname{hypergeom}_{2,1}\left(\left[1,\frac{\beta-t}{\beta}\right],\left[\frac{2\beta-t}{\beta}\right],1-p\right),</math>

: <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 hypergeom_2,1 is hypergeometric function. This function

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

is also known as ''Barnes's extended hypergeometric function''. The

definition of <math>F_{p,q}({n,d},\lambda)</math> is

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

: <math>F_{p,q}({n,d},\lambda)=\sum_{k=0}^\infty \frac{\lambda^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>

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

:<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:<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>

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

where <math>{n}=[n_1, n_2, ..., n_p]</math>, <math>p</math> is the number of

are given, respectively, by

operands of <math>{n}</math>, <math>{d}=[d_1, d_2, \dots, d_q]</math> and <math>q</math> is

:<math>E(X)=-\frac{\operatorname{Li}_2(1-p)}{\beta\ln p},</math>

the number of operands of <math>{d}</math>. Generalized hypergeometric

function is quickly evaluated and readily available in standard

software such as Maple.

:<math>\operatorname{Var}(X)=-\frac{2 \operatorname{Li}_3(1-p)}{\beta^2\ln p}-\left(\frac{ \operatorname{Li}_2(1-p)}{\beta\ln p}\right)^2.</math>

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>

<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 polylogarithm function and it 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 ===

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

The survival function (also known as reliability

function) andThe hazard[[survival function]] (also known as failurethe ratereliability

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

function) of the EL distribution are given, respectively, by

Line 119 ⟶ 85:

The mean residual lifetime of the EL distribution is given by

: <math>m(x_0;p,\beta)=E(X-x_0|X\geq x_0;\beta,p)=-\frac{\operatorname{dilogLi}_2(1-(1-p)e^{-\beta x_0})}{\beta \ln(1-(1-p)e^{-\beta x_0})}</math>

where dilog<math>\operatorname{Li}_2</math> 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;β'':

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

== 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).<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>

Line 139 ⟶ 103:

: <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]]