Matrix-exponential distribution

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

Matrix-exponential

Parameters	α, T, s
Support	x ∈ [0, ∞)
PDF	α ex Ts
CDF	1 + αexTT−1s

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

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

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

Moments

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

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

Fitting

Matrix exponential distributions can be fitted using maximum likelihood estimation.

Software

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

See also

• Rational arrival process