# 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]

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

(and 0 when x < 0) where

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

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]

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

## Fitting

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