Matrix-exponential distribution

Short description: Absolutely continuous distribution with rational Laplace–Stieltjes transform

Matrix-exponentialParameters|  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 f(x)=αexT𝐬 for x≥0 (and 0 when x < 0), and the cumulative distribution function is F(t)=1−αeAt1[3] where 1 is a vector of 1s and

α∈ℝ1×n,T∈ℝn×n,s∈ℝn×1.

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.

## Contents

* 1 Moments
* 2 Fitting
* 3 Software
* 4 See also
* 5 References

## Moments

If X has a matrix-exponential distribution then the kth moment is given by[2]

E⁡(Xk)=(−1)k+1k!αT−(k+1)𝐬.

## Fitting

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

## Software

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

## See also

* Rational arrival process

## References

1. ↑ 1.0 1.1 Asmussen, S. R.; o’Cinneide, C. A. (2006). "Matrix-Exponential Distributions". Encyclopedia of Statistical Sciences. doi:10.1002/0471667196.ess1092.pub2. ISBN 0471667196.
2. ↑ 2.0 2.1 2.2 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.ph) — butools 2.0 documentation". http://webspn.hit.bme.hu/~telek/tools/butools/doc/ph.html.
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.

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