Matrix-exponential distribution

In probability theory, the matrix-exponential distribution is an absolutely continuous distribution with rational Laplace–Stieltjes transform. They were introduced by David Cox in 1955 as distributions with rational Laplace–Stieltjes transforms. The probability density function is f ( x ) = α e x T s for x ≥ 0 {\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 F ( t ) = 1 − α e A t 1 {\displaystyle F(t)=1-\alpha e^{{\textbf {A}}t}{\textbf {1}}} where 1 is a vector of 1s and

α ∈ R 1 × n , T ∈ R n × n , s ∈ R n × 1 . {\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. 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

E ⁡ ( X k ) = ( − 1 ) k + 1 k ! α T − ( k + 1 ) s . {\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.

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

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