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Exponential-logarithmic distribution
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<h2 id="introduction">Introduction</h2> <p>The study of lengths of the lives of organisms, devices, materials, etc., is of major importance in the <a href="/facts/Biological/1M1ee3pB">biological</a> and <a href="/facts/Engineering/RFVg8H3I">engineering</a> 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). </p><p>The exponential-logarithmic model, together with its various properties, are studied by Tahmasbi and Rezaei (2008).<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> This model is obtained under the concept of population heterogeneity (through the process of compounding). </p> <h2 id="properties-of-the-distribution">Properties of the distribution</h2> <h3>Distribution</h3> <p>The <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> (pdf) of the EL distribution is given by Tahmasbi and Rezaei (2008)<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> f ( x ; p , β ) := ( 1 − ln p ) β ( 1 − p ) e − β x 1 − ( 1 − p ) e − β x {\displaystyle f(x;p,\beta ):=\left({\frac {1}{-\ln p}}\right){\frac {\beta (1-p)e^{-\beta x}}{1-(1-p)e^{-\beta x}}}} <p>where p ∈ ( 0 , 1 ) {\displaystyle p\in (0,1)} and β > 0 {\displaystyle \beta >0} . This function is strictly decreasing in x {\displaystyle x} and tends to zero as x → ∞ {\displaystyle x\rightarrow \infty } . The EL distribution has its <a href="/facts/Mode_(statistics)/vq5tGp0G">modal value</a> of the density at x=0, given by </p> β ( 1 − p ) − p ln p {\displaystyle {\frac {\beta (1-p)}{-p\ln p}}} <p>The EL reduces to the <a href="/facts/Exponential_distribution/yY3EdV0u">exponential distribution</a> with rate parameter β {\displaystyle \beta } , as p → 1 {\displaystyle p\rightarrow 1} . </p><p>The <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> is given by </p> F ( x ; p , β ) = 1 − ln ( 1 − ( 1 − p ) e − β x ) ln p , {\displaystyle F(x;p,\beta )=1-{\frac {\ln(1-(1-p)e^{-\beta x})}{\ln p}},} <p>and hence, the <a href="/facts/Median/YwXGWTyr">median</a> is given by </p> x median = ln ( 1 + p ) β {\displaystyle x_{\text{median}}={\frac {\ln(1+{\sqrt {p}})}{\beta }}} . <h3>Moments</h3> <p>The <a href="/facts/Moment_generating_function/eTmk43QU">moment generating function</a> of X {\displaystyle X} can be determined from the pdf by direct integration and is given by </p> M X ( t ) = E ( e t X ) = − β ( 1 − p ) ln p ( β − t ) F 2 , 1 ( [ 1 , β − t β ] , [ 2 β − t β ] , 1 − p ) , {\displaystyle 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),} <p>where F 2 , 1 {\displaystyle F_{2,1}} is a <a href="/facts/Hypergeometric_function/yTEC6NBJ">hypergeometric function</a>. This function is also known as <i>Barnes's extended hypergeometric function</i>. The definition of F N , D ( n , d , z ) {\displaystyle F_{N,D}({n,d},z)} is </p> F N , D ( n , d , z ) := ∑ k = 0 ∞ z k ∏ i = 1 p Γ ( n i + k ) Γ − 1 ( n i ) Γ ( k + 1 ) ∏ i = 1 q Γ ( d i + k ) Γ − 1 ( d i ) {\displaystyle 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})}}} <p>where n = [ n 1 , n 2 , … , n N ] {\displaystyle n=[n_{1},n_{2},\dots ,n_{N}]} and d = [ d 1 , d 2 , … , d D ] {\displaystyle {d}=[d_{1},d_{2},\dots ,d_{D}]} . </p><p>The moments of X {\displaystyle X} can be derived from M X ( t ) {\displaystyle M_{X}(t)} . For r ∈ N {\displaystyle r\in \mathbb {N} } , the raw moments are given by </p> E ( X r ; p , β ) = − r ! Li r + 1 ( 1 − p ) β r ln p , {\displaystyle E(X^{r};p,\beta )=-r!{\frac {\operatorname {Li} _{r+1}(1-p)}{\beta ^{r}\ln p}},} <p>where Li a ( z ) {\displaystyle \operatorname {Li} _{a}(z)} is the <a href="/facts/Polylogarithm/CKrkJMKn">polylogarithm</a> function which is defined as follows:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> Li a ( z ) = ∑ k = 1 ∞ z k k a . {\displaystyle \operatorname {Li} _{a}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{a}}}.} <p>Hence the <a href="/facts/Mean/swcEd4Pg">mean</a> and <a href="/facts/Variance/ULBJKXD1">variance</a> of the EL distribution are given, respectively, by </p> E ( X ) = − Li 2 ( 1 − p ) β ln p , {\displaystyle E(X)=-{\frac {\operatorname {Li} _{2}(1-p)}{\beta \ln p}},} Var ( X ) = − 2 Li 3 ( 1 − p ) β 2 ln p − ( Li 2 ( 1 − p ) β ln p ) 2 . {\displaystyle \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}.} <h3>The survival, hazard and mean residual life functions</h3> <p>The <a href="/facts/Survival_function/jkdCmDKq">survival function</a> (also known as the reliability function) and <a href="/facts/Hazard_function/shlTIKKg">hazard function</a> (also known as the failure rate function) of the EL distribution are given, respectively, by </p> s ( x ) = ln ( 1 − ( 1 − p ) e − β x ) ln p , {\displaystyle s(x)={\frac {\ln(1-(1-p)e^{-\beta x})}{\ln p}},} h ( x ) = − β ( 1 − p ) e − β x ( 1 − ( 1 − p ) e − β x ) ln ( 1 − ( 1 − p ) e − β x ) . {\displaystyle h(x)={\frac {-\beta (1-p)e^{-\beta x}}{(1-(1-p)e^{-\beta x})\ln(1-(1-p)e^{-\beta x})}}.} <p>The mean residual lifetime of the EL distribution is given by </p> m ( x 0 ; p , β ) = E ( X − x 0 | X ≥ x 0 ; β , p ) = − Li 2 ( 1 − ( 1 − p ) e − β x 0 ) β ln ( 1 − ( 1 − p ) e − β x 0 ) {\displaystyle m(x_{0};p,\beta )=E(X-x_{0}|X\geq x_{0};\beta ,p)=-{\frac {\operatorname {Li} _{2}(1-(1-p)e^{-\beta x_{0}})}{\beta \ln(1-(1-p)e^{-\beta x_{0}})}}} <p>where Li 2 {\displaystyle \operatorname {Li} _{2}} is the <a href="/facts/Dilogarithm/lsnCj3Gb">dilogarithm</a> function </p> <h3>Random number generation</h3> <p>Let <i>U</i> be a <a href="/facts/Random_variate/dTkjAeaj">random variate</a> from the standard <a href="/facts/Uniform_distribution_(continuous)/XbnlVljT">uniform distribution</a>. Then the following transformation of <i>U</i> has the EL distribution with parameters <i>p</i> and <i>β</i>: </p> X = 1 β ln ( 1 − p 1 − p U ) . {\displaystyle X={\frac {1}{\beta }}\ln \left({\frac {1-p}{1-p^{U}}}\right).} <h2 id="estimation-of-the-parameters">Estimation of the parameters</h2> <p>To estimate the parameters, the <a href="/facts/Expectation-maximization_algorithm/32P2bdBc">EM algorithm</a> is used. This method is discussed by Tahmasbi and Rezaei (2008).<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> The EM iteration is given by </p> β ( h + 1 ) = n ( ∑ i = 1 n x i 1 − ( 1 − p ( h ) ) e − β ( h ) x i ) − 1 , {\displaystyle \beta ^{(h+1)}=n\left(\sum _{i=1}^{n}{\frac {x_{i}}{1-(1-p^{(h)})e^{-\beta ^{(h)}x_{i}}}}\right)^{-1},} p ( h + 1 ) = − n ( 1 − p ( h + 1 ) ) ln ( p ( h + 1 ) ) ∑ i = 1 n { 1 − ( 1 − p ( h ) ) e − β ( h ) x i } − 1 . {\displaystyle 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}}}.} <h2 id="related-distributions">Related distributions</h2> <p>The EL distribution has been generalized to form the Weibull-logarithmic distribution.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>If <i>X</i> is defined to be the <a href="/facts/Random_variable/TwTBXnLT">random variable</a> which is the minimum of <i>N</i> independent realisations from an <a href="/facts/Exponential_distribution/yY3EdV0u">exponential distribution</a> with rate parameter <i>β</i>, and if <i>N</i> is a realisation from a <a href="/facts/Logarithmic_distribution/4vtljgLE">logarithmic distribution</a> (where the parameter <i>p</i> in the usual parameterisation is replaced by (1 − <i>p</i>)), then <i>X</i> has the exponential-logarithmic distribution in the parameterisation used above. </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>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 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Lewin, L. (1981) Polylogarithms and Associated Functions, North Holland, Amsterdam. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>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 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Ciumara, Roxana; Preda, Vasile (2009) "The Weibull-logarithmic distribution in lifetime analysis and its properties". In: L. Sakalauskas, C. Skiadas and E. K. Zavadskas (Eds.) Applied Stochastic Models and Data Analysis Archived 2011-05-18 at the Wayback Machine, The XIII International Conference, Selected papers. Vilnius, 2009 ISBN 978-9955-28-463-5 <a href="https://www.proquest.com/openview/7f1efa684243ce36231867620f09373a/1" target="_blank">https://www.proquest.com/openview/7f1efa684243ce36231867620f09373a/1</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>