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{{Short description|Family of lifetime distributions with decreasing failure rate}}
{{Infobox probability distribution
In [[probability theory]] and [[statistics]], the '''exponential-logarithmic (EL) distribution''' is a family of lifetime [[probability distribution|distributions]] with▼
| type = continuous
decreasing [[failure rate]], defined on the interval (0, ∞). This distribution is [[Parametric family|parameterized]] by two parameters <math>p\in(0,1)</math> and <math>\beta >0</math>.▼
| cdf_image =
| notation =
| parameters = <math>p\in (0,1)</math><br><math>\beta >0</math>
▲ <CAPTION>Exponential-Logarithmic distribution (EL)</CAPTION>
| mode = 0
| variance
| skewness =
| kurtosis =
| entropy =
▲ <TD><math>x\in(0,\infty)</math></TD></TR>
| mgf
| cf =
▲ <TH>Probability density function (pdf)</TH>
| pgf =
| fisher =
}}
▲In [[probability theory]] and [[statistics]], the '''
▲decreasing [[failure rate]], defined on the interval
▲ <TD><math>1-\frac{\ln(1-(1-p) e^{-\beta x})}{\ln p}</math></TD></TR>
▲ <TD><math>-\frac{\text{polylog}(2,1-p)}{\beta\ln p}</math></TD></TR>
▲ <TD><math>\frac{\ln(1+\sqrt{p})}{\beta}</math></TD></TR>
▲ <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>
== 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
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'',
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"/>
▲:<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>
:<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 [[
:<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
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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Let ''U'' be a [[random variate]] from the standard [[Uniform distribution (continuous)|uniform distribution]].
Then the following transformation of ''U'' has the EL distribution with
parameters ''p'' and ''
: <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 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 ''β'', and if ''N'' is a realisation from a [[logarithmic distribution]] (where the parameter ''p'' in the usual parameterisation is replaced by {{nowrap|1=(1 − ''p'')}}), then ''X'' has the exponential-logarithmic distribution in the parameterisation used above.
==References==
{{Reflist}}
{{ProbDistributions|continuous-semi-infinite}}
[[Category:Continuous distributions]]
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