Matrix-exponential distribution explained
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
F(t)=1-\alphaebf{At}bf{1}
[3] where
1 is a vector of 1s and
\begin{align}
\alpha&\inR1 x ,\\
T&\inRn x ,\\
s&\inRn x .
\end{align}
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. 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
\operatornameE(Xk)=(-1)k+1k!\alphaT-(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
Notes and References
- Book: 10.1002/0471667196.ess1092.pub2. Matrix-Exponential Distributions. Encyclopedia of Statistical Sciences. 2006. Asmussen . S. R. . o’Cinneide . C. A. . 0471667196.
- 10.1080/15326340802232186. Characterization of Matrix-Exponential Distributions. Stochastic Models. 24. 3. 339. 2008. Bean . N. G. . Fackrell . M. . Taylor . P. .
- Web site: Tools for Phase-Type Distributions (butools.ph) — butools 2.0 documentation . 2022-04-16 . webspn.hit.bme.hu.
- 10.1239/aap/1175266478. On matrix exponential distributions. Advances in Applied Probability. 39. 271–292. 2007. He . Q. M. . Zhang . H. . Applied Probability Trust. free.
- 10.1081/STM-200056227. Fitting with Matrix-Exponential Distributions. Stochastic Models. 21. 2–3. 377. 2005. Fackrell . M. .