# Matrix-exponential distribution

Parameters α, T, s x ∈ [0, ∞) α ex Ts 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]

${\displaystyle f(x)=\mathbf {\alpha } e^{x\,T}\mathbf {s} {\text{ for }}x\geq 0}$

(and 0 when x < 0) where

{\displaystyle {\begin{aligned}\alpha &\in \mathbb {R} ^{1\times n},\\T&\in \mathbb {R} ^{n\times n},\\s&\in \mathbb {R} ^{n\times 1}.\end{aligned}}}

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]

${\displaystyle \operatorname {E} (X^{k})=(-1)^{k+1}k!\mathbf {\alpha } T^{-(k+1)}\mathbf {s} .}$

## Fitting

Matrix exponential distributions can be fitted using maximum likelihood estimation.[4]

## References

1. ^ a b Asmussen, S. R.; o’Cinneide, C. A. (2006). "Matrix-Exponential Distributions". Encyclopedia of Statistical Sciences. doi:10.1002/0471667196.ess1092.pub2. ISBN 0471667196.
2. ^ a b c Bean, N. G.; Fackrell, M.; Taylor, P. (2008). "Characterization of Matrix-Exponential Distributions". Stochastic Models. 24 (3): 339. doi:10.1080/15326340802232186.
3. ^ 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.
4. ^ Fackrell, M. (2005). "Fitting with Matrix-Exponential Distributions". Stochastic Models. 21 (2–3): 377. doi:10.1081/STM-200056227.