Matrix-exponential distribution
Absolutely continuous distribution with rational Laplace–Stieltjes transform From Wikipedia, the free encyclopedia
In probability theory, the matrix-exponential distribution is an absolutely continuous distribution with rational Laplace–Stieltjes transform.[1] They were first introduced by David Cox in 1955 as distributions with rational Laplace–Stieltjes transforms.[2]
Parameters | α, T, s | ||
---|---|---|---|
Support | x ∈ [0, ∞) | ||
α ex Ts | |||
CDF | 1 + αexTT−1s |
The probability density function is (and 0 when x < 0), and the cumulative distribution function is [3] where 1 is a vector of 1s and
There are no restrictions on the parameters α, T, s other than that they correspond to a probability distribution.[4] There is no straightforward way to ascertain if a particular set of parameters form such a distribution.[2] The dimension of the matrix T is the order of the matrix-exponential representation.[1]
The distribution is a generalisation of the phase-type distribution.
Moments
If X has a matrix-exponential distribution then the kth moment is given by[2]
Fitting
Matrix exponential distributions can be fitted using maximum likelihood estimation.[5]
Software
- BuTools a MATLAB and Mathematica script for fitting matrix-exponential distributions to three specified moments.
See also
References
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