Matrix-exponential distribution

Matrix-exponential

Parameters	α, T, s
Support	x ∈ [0, ∞)
PDF	α e^x Ts
CDF	1 + αe^xTT^−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]

The probability density function is

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

(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]

Software

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

See also

• Rational arrival process

References

1. Asmussen, S. R.; o’Cinneide, C. A. (2006). "Matrix-Exponential Distributions". Encyclopedia of Statistical Sciences. doi:10.1002/0471667196.ess1092.pub2. ISBN 0471667196.
2. Bean, N. G.; Fackrell, M.; Taylor, P. (2008). "Characterization of Matrix-Exponential Distributions". Stochastic Models 24 (3): 339. doi:10.1080/15326340802232186.
3. He, Q. M.; Zhang, H. (2007). "On matrix exponential distributions". Advances in Applied Probability (Applied Probability Trust) 39: 271–292. doi:10.1239/aap/1175266478.
4. Fackrell, M. (2005). "Fitting with Matrix-Exponential Distributions". Stochastic Models 21 (2–3): 377. doi:10.1081/STM-200056227.