Parameters α, T, s
Support x ∈ [0, ∞)
PDF α ex Ts
CDF 1 + αexTT−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 (and 0 when x < 0), and the cumulative distribution function is [3] where 1 is a vector of 1s and

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 by[2]


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


See also


  1. ^ a b Asmussen, S. R.; o’Cinneide, C. A. (2006). "Matrix-Exponential Distributions". Encyclopedia of Statistical Sciences. doi:10.1002/0471667196.ess1092.pub2. ISBN 0471667196.
  2. ^ a b c Bean, N. G.; Fackrell, M.; Taylor, P. (2008). "Characterization of Matrix-Exponential Distributions". Stochastic Models. 24 (3): 339. doi:10.1080/15326340802232186.
  3. ^ "Tools for Phase-Type Distributions ( — butools 2.0 documentation". Retrieved 2022-04-16.
  4. ^ 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.
  5. ^ Fackrell, M. (2005). "Fitting with Matrix-Exponential Distributions". Stochastic Models. 21 (2–3): 377. doi:10.1081/STM-200056227.