Matrix-exponential distribution
Parameters | α, T, s |
---|---|
Support | x ∈ [0, ∞) |
α ex Ts | |
CDF | 1 + αexTT−1s |
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]
The probability density function is
(and 0 when x < 0) where
There are no restrictions on the parameters α, T, s other than that they correspond to a probability distribution.[3] 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.[4]
Software
- BuTools a MATLAB and Mathematica script for fitting matrix-exponential distributions to three specified moments.
See also
References
- 1 2 Asmussen, S. R.; o’Cinneide, C. A. (2006). "Matrix-Exponential Distributions". Encyclopedia of Statistical Sciences. doi:10.1002/0471667196.ess1092.pub2. ISBN 0471667196.
- 1 2 3 Bean, N. G.; Fackrell, M.; Taylor, P. (2008). "Characterization of Matrix-Exponential Distributions". Stochastic Models. 24 (3): 339. doi:10.1080/15326340802232186.
- ↑ He, Q. M.; Zhang, H. (2007). "On matrix exponential distributions". Advances in Applied Probability. Applied Probability Trust. 39: 271–292. doi:10.1239/aap/1175266478.
- ↑ Fackrell, M. (2005). "Fitting with Matrix-Exponential Distributions". Stochastic Models. 21 (2–3): 377. doi:10.1081/STM-200056227.