Matrix-exponential distribution

In probability theory, the matrix-exponential distribution is an absolutely continuous distribution with rational Laplace–Stieltjes transform.¹ They were introduced by David Cox in 1955 as distributions with rational Laplace–Stieltjes transforms.²

The probability density function is ${\displaystyle f(x)=\mathbf {\alpha } e^{x\,T}\mathbf {s} {\text{ for }}x\geq 0}$ (and 0 when x < 0), and the cumulative distribution function is ${\displaystyle F(t)=1-\alpha e^{{\textbf {A}}t}{\textbf {1}}}$³ where 1 is a vector of 1s and

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

There are no restrictions on the parameters α, T, s other than that they correspond to a probability distribution.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.

If X has a matrix-exponential distribution then the kth moment is given by²

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

Matrix exponential distributions can be fitted using maximum likelihood estimation.⁵

• BuTools a MATLAB and Mathematica script for fitting matrix-exponential distributions to three specified moments.