Exponential-logarithmic distribution: Difference between revisions

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=== Distribution ===

The [[probability density function]] (pdf) of the EL distribution is given by

:<math>F_X f(x; p, \beta) := \left( \frac{1}{-\ln p}\right) \frac{\ln(1-beta(1-p) e^{-\beta x})}{1-(1-p)e^{\lnbeta px},} </math>

distribution is monotone decreasing with

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 [[mode|modal value]] given, at x=0, by

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

The EL reduces to the [[exponential distribution]] with parameter <math>\beta</math>, as <math>p\rightarrow 1</math>.

The [[cumulative distribution function]] is given by

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

:<math>F(x;p,\mathrm{median}beta)=1-\frac{\ln(1+\sqrt{-(1-p) e^{-\beta x})}{\betaln p},</math>.

<math>x</math> and tending to zero as <math>x\rightarrow \infty</math>. The EL leads to the

and hence, the [[median]] is given by

exponential distribution with parameter <math>\beta</math>, as <math>p\rightarrow 1</math>.

:<math>x_\text{median}=\frac{\ln(1+\sqrt{p})}{\beta}</math>.

The distribution function is given by

:<math>F_X(x;p,\beta)=1-\frac{\ln(1-(1-p) e^{-\beta x})}{\ln p},</math>

and hence, the median is given by

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

=== Moments ===

The [[moment generating function]] of <math>X</math> can 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)} \operatorname{hypergeom}_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_{pN,qD}({n,d},\lambdaz):=\sum_{k=0}^\infty \frac{\lambda 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_pn_N]</math>, and <math>p{d}=[d_1, d_2, \dots , d_D]</math>. is the number of

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

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

: <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>

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

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

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

function is quickly evaluated and readily available in standard

software such as Maple.

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

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

:<math>E(X^r;p,\beta)=-r!\frac{r! polylog(\operatorname{Li}_{r+1,}(1-p) }{\beta^r\ln p}, r\in\mathbb{N},</math>

where <math>polylog\operatorname{Li}_a(.z)</math> is the [[polylogarithm]] function which is defined as

follows (Lewin, 1981) <ref>Lewin, L., 1981, Polylogarithms and Associated Functions, North

Holland, Amsterdam.</ref>

:<math>polylog\operatorname{Li}_a(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

:<math>E(X)=-\frac{polylog\operatorname{Li}_2(2,1-p)}{\beta\ln p},</math>

:<math>\operatorname{Var}(X)=-\frac{2 polylog\operatorname{Li}_3(3,1-p)}{\beta^2\ln p}-\left(\frac{ polylog^2\operatorname{Li}_2(2,1-p)}{\beta^2\ln^2 p}\right)^2.</math>

=== The survival, hazard and mean residual life functions ===

The [[survival function]] (also known as the reliability

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

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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 is the [[dilogarithm]] function defined as follows:

:where <math>\operatorname{dilogLi}(a)=\int_1^a \frac{\ln(x)}{1-x} \, dx._2</math> is the [[dilogarithm]] function

=== Random number generation ===