Matrix-exponential distribution
| Parameters | α, T, s | ||
|---|---|---|---|
| Support | x ∈ [0, ∞) | ||
| 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 [math]\displaystyle{ f(x) = \mathbf{\alpha} e^{x\,T} \mathbf{s} \text{ for }x\ge 0 }[/math] (and 0 when x < 0), and the cumulative distribution function is [math]\displaystyle{ F(t) = 1 - \alpha e^{\textbf{A}t} \textbf{1} }[/math][3] where 1 is a vector of 1s and
- [math]\displaystyle{ \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} }[/math]
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.
Moments
If X has a matrix-exponential distribution then the kth moment is given by[2]
- [math]\displaystyle{ \operatorname E(X^k) = (-1)^{k+1}k! \mathbf{\alpha} T^{-(k+1)}\mathbf{s}. }[/math]
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
References
- ↑ 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.
- ↑ 2.0 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.
- ↑ "Tools for Phase-Type Distributions (butools.ph) — butools 2.0 documentation". http://webspn.hit.bme.hu/~telek/tools/butools/doc/ph.html.
- ↑ 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.
- ↑ Fackrell, M. (2005). "Fitting with Matrix-Exponential Distributions". Stochastic Models 21 (2–3): 377. doi:10.1081/STM-200056227.
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