Exponential-logarithmic distribution: Difference between revisions

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

{{Orphan|date=August 2009}}

{{Infobox probability distribution

| name = Exponential-Logarithmic distribution (EL)

{{Wikify|date=August 2009}}

| type = continuous

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

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

| cdf_image =

is a family of lifetime distribution with<br>

| notation =

decreasing failure rate, defined on the interval <math>(0,\infty)</math>. This distribution is parameterized by two

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

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

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

<TABLE class="infobox bordered wikitable"

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

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

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

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

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

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

| mode = 0

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

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

| skewness =

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

| kurtosis =

<TR vAlign=top>

| entropy =

<TH>Parameters</TH>

| mgf <TD><SPAN> = <math>-\frac{\beta(1-p)}{\inln p (0\beta-t)} \text{hypergeom}_{2,1)} </math></SPAN><BR><SPANbr> <math>([1,\frac{\beta >0-t}{\beta}],[\frac{2\beta-t}{\beta}],1-p)</math></SPAN></TD></TR>

| cf =

<TR>

| pgf =

<TH>Support</TH>

| fisher =

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

}}

<TR>

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

<TH>Probability density function (pdf)</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>\frac{1}{-\ln p} \times \frac{\beta(1-p) e^{-\beta

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

<TR>

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

<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]] 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 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).

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'', 52 (8), 3889-3901. {{doi|10.1016/j.csda.2007.12.002}}</ref>

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).

== 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 129 ⟶ 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 149 ⟶ 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:Probability theory]]

[[Category:Continuous distributions]]

[[Category:Survival analysis]]