Exponential-Logarithmic distribution (EL) | ||||||||||||||||||||||||||||||||
Type: | continuous | |||||||||||||||||||||||||||||||
Parameters: | p\in(0,1) \beta>0 | |||||||||||||||||||||||||||||||
Support: | x\in[0,infty)
x
hypergeom2,1
],1-p) |
In probability theory and statistics, the Exponential-Logarithmic (EL) distribution is a family of lifetime distributions withdecreasing failure rate, defined on the interval [0, ∞). This distribution is [[Parametric family|parameterized]] by two parameters
p\in(0,1)
\beta>0
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).[1] This model is obtained under the concept of population heterogeneity (through the process ofcompounding).
The probability density function (pdf) of the EL distribution is given by Tahmasbi and Rezaei (2008)[1]
f(x;p,\beta):=\left(
1 | |
-lnp |
\right)
\beta(1-p)e-\beta | |
1-(1-p)e-\beta |
p\in(0,1)
\beta>0
x
x → infty
\beta(1-p) | |
-plnp |
\beta
p → 1
The cumulative distribution function is given by
F(x;p,\beta)=1- | ln(1-(1-p)e-\beta) |
lnp |
,
x | ||||
|
The moment generating function of
X
MX(t)=E(etX)=-
\beta(1-p) | |
lnp(\beta-t) |
F2,1\left(\left[1,
\beta-t | \right],\left[ | |
\beta |
2\beta-t | |
\beta |
\right],1-p\right),
where
F2,1
FN,D({n,d},z)
FN,D
infty | |
(n,d,z):=\sum | |
k=0 |
| |||||||||
|
n=[n1,n2,...,nN]
{d}=[d1,d2,...,dD]
The moments of
X
MX(t)
r\inN
| ||||
E(X |
(1-p)}{\betarlnp},
\operatorname{Li}a(z)
\operatorname{Li}a(z)
infty | |
=\sum | |
k=1 |
zk | |
ka |
.
Hence the mean and variance of the EL distributionare given, respectively, by
E(X)=- | \operatorname{Li |
2(1-p)}{\betaln |
p},
\operatorname{Var}(X)=- | 2\operatorname{Li |
3(1-p)}{\beta |
2lnp}-\left(
\operatorname{Li | |
2(1-p)}{\betaln |
p}\right)2.
The survival function (also known as the reliabilityfunction) and hazard function (also known as the failure ratefunction) of the EL distribution are given, respectively, by
s(x)= | ln(1-(1-p)e-\beta) |
lnp |
,
h(x)= | -\beta(1-p)e-\beta |
(1-(1-p)e-\beta)ln(1-(1-p)e-\beta) |
.
The mean residual lifetime of the EL distribution is given by
m(x0;p,\beta)=E(X-x0|X\geq
-\betax0 | ||||
x | ||||
|
)}{\beta
-\betax0 | |
ln(1-(1-p)e |
)}
where
\operatorname{Li}2
Let U be a random variate from the standard uniform distribution.Then the following transformation of U has the EL distribution withparameters p and β:
X=
1 | |
\beta |
ln\left(
1-p | |
1-pU |
\right).
To estimate the parameters, the EM algorithm is used. This method is discussed by Tahmasbi and Rezaei (2008).[1] The EM iteration is given by
\beta(h+1)=n\left(
| ||||||||||||||
\sum | ||||||||||||||
i=1 |
\right)-1,
p(h+1)=
-n(1-p(h+1)) | ||||||||||||
|
-1
The EL distribution has been generalized to form the Weibull-logarithmic distribution.[3]
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), then X has the exponential-logarithmic distribution in the parameterisation used above.