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
f(x)=\alphaexTsforx\ge0
F(t)=1-\alphaebf{At}bf{1}
\begin{align} \alpha&\inR1 x ,\\ T&\inRn x ,\\ s&\inRn x . \end{align}
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. The dimension of the matrix T is the order of the matrix-exponential representation.
The distribution is a generalisation of the phase-type distribution.
If X has a matrix-exponential distribution then the kth moment is given by
\operatornameE(Xk)=(-1)k+1k!\alphaT-(k+1)s.
Matrix exponential distributions can be fitted using maximum likelihood estimation.[5]