Exponential-logarithmic distribution

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).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}}

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

compounding).

The probability density function (pdf) of the EL distribution is given by Tahmasbi and Rezaei (2008)

:$f(x;\; p,\; \backslash beta)\; :=\; \backslash left(\; \backslash frac\{1\}\{-\backslash ln\; p\}\backslash right)\; \backslash frac\{\backslash beta(1-p)e^\{-\backslash beta\; x\}\}\{1-(1-p)e^\{-\backslash beta\; x\}\}$

where $p\backslash in\; (0,1)$ and $\backslash beta\; >0$. This function is strictly decreasing in $x$ and tends to zero as $x\backslash rightarrow\; \backslash infty$. The EL distribution has its modal value of the density at x=0, given by

:$\backslash frac\{\backslash beta\; (1-p)\}\{-p\; \backslash ln\; p\}$

The EL reduces to the exponential distribution with rate parameter $\backslash beta$, as $p\backslash rightarrow\; 1$.

The cumulative distribution function is given by

:$F(x;p,\backslash beta)=1-\backslash frac\{\backslash ln(1-(1-p)\; e^\{-\backslash beta\; x\})\}\{\backslash ln\; p\},$

and hence, the median is given by

:$x\_\backslash text\{median\}=\backslash frac\{\backslash ln(1+\backslash sqrt\{p\})\}\{\backslash beta\}$.

The moment generating function of $X$ can be determined from the pdf by direct integration and is given by

: $M\_X(t)\; =\; E(e^\{tX\})\; =\; -\backslash frac\{\backslash beta(1-p)\}\{\backslash ln\; p\; (\backslash beta-t)\}\; F\_\{2,1\}\backslash left(\backslash left[1,\backslash frac\{\backslash beta-t\}\{\backslash beta\}\backslash right],\backslash left[\backslash frac\{2\backslash beta-t\}\{\backslash beta\}\backslash right],1-p\backslash right),$

where $F\_\{2,1\}$ is a hypergeometric function. This function is also known as Barnes's extended hypergeometric function. The definition of $F\_\{N,D\}(\{n,d\},z)$ is

: $F\_\{N,D\}(n,d,z):=\backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{\; z^k\; \backslash prod\_\{i=1\}^p\backslash Gamma(n\_i+k)\backslash Gamma^\{-1\}(n\_i)\}\{\backslash Gamma(k+1)\backslash prod\_\{i=1\}^q\backslash Gamma(d\_i+k)\backslash Gamma^\{-1\}(d\_i)\}$

where $n=[n\_1,\; n\_2,\backslash dots\; ,\; n\_N]$ and $\{d\}=[d\_1,\; d\_2,\; \backslash dots\; ,\; d\_D]$.

The moments of $X$ can be derived from $M\_X(t)$. For

$r\backslash in\backslash mathbb\{N\}$, the raw moments are given by

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

where $\backslash operatorname\{Li\}\_a(z)$ is the polylogarithm function which is defined as

follows:Lewin, L. (1981) Polylogarithms and Associated Functions, North

Holland, Amsterdam.

:$\backslash operatorname\{Li\}\_a(z)\; =\backslash sum\_\{k=1\}^\{\backslash infty\}\backslash frac\{z^k\}\{k^a\}.$

Hence the mean and variance of the EL distribution

are given, respectively, by

:$E(X)=-\backslash frac\{\backslash operatorname\{Li\}\_2(1-p)\}\{\backslash beta\backslash ln\; p\},$

:$\backslash operatorname\{Var\}(X)=-\backslash frac\{2\; \backslash operatorname\{Li\}\_3(1-p)\}\{\backslash beta^2\backslash ln\; p\}-\backslash left(\backslash frac\{\; \backslash operatorname\{Li\}\_2(1-p)\}\{\backslash beta\backslash ln\; p\}\backslash right)^2.$

File:Hazard EL.png

The survival function (also known as the reliability

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

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

: $s(x)=\backslash frac\{\backslash ln(1-(1-p)e^\{-\backslash beta\; x\})\}\{\backslash ln\; p\},$

: $h(x)=\backslash frac\{-\backslash beta(1-p)e^\{-\backslash beta\; x\}\}\{(1-(1-p)e^\{-\backslash beta\; x\})\backslash ln(1-(1-p)e^\{-\backslash beta\; x\})\}.$

The mean residual lifetime of the EL distribution is given by

: $m(x\_0;p,\backslash beta)=E(X-x\_0|X\backslash geq\; x\_0;\backslash beta,p)=-\backslash frac\{\backslash operatorname\{Li\}\_2(1-(1-p)e^\{-\backslash beta\; x\_0\})\}\{\backslash beta\; \backslash ln(1-(1-p)e^\{-\backslash beta\; x\_0\})\}$

where $\backslash operatorname\{Li\}\_2$ is the dilogarithm function

Let U be a random variate from the standard uniform distribution.

Then the following transformation of U has the EL distribution with

parameters p and β:

: $X\; =\; \backslash frac\{1\}\{\backslash beta\}\backslash ln\; \backslash left(\backslash frac\{1-p\}\{1-p^U\}\backslash right).$

To estimate the parameters, the EM algorithm is used. This method is discussed by Tahmasbi and Rezaei (2008). The EM iteration is given by

: $\backslash beta^\{(h+1)\}\; =\; n\; \backslash left(\; \backslash sum\_\{i=1\}^n\backslash frac\{x\_i\}\{1-(1-p^\{(h)\})e^\{-\backslash beta^\{(h)\}x\_i\}\}\; \backslash right)^\{-1\},$

: $p^\{(h+1)\}=\backslash frac\{-n(1-p^\{(h+1)\})\}\; \{\; \backslash ln(\; p^\{(h+1)\})\; \backslash sum\_\{i=1\}^n$

\{1-(1-p^{(h)})e^{-\beta^{(h)} x_i}\}^{-1}}.

The EL distribution has been generalized to form the Weibull-logarithmic distribution.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}}

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.

{{Reflist}}

{{ProbDistributions|continuous-semi-infinite}}

Category:Continuous distributions

Category:Survival analysis