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 (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.
Moments
If X has a matrix-exponential distribution then the kth moment is given by[2]
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
- ^ 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.
- ^ 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.
- ^ "Tools for Phase-Type Distributions (butools.ph) — butools 2.0 documentation". webspn.hit.bme.hu. Retrieved 2022-04-16.
- ^ He, Q. M.; Zhang, H. (2007). "On matrix exponential distributions". Advances in Applied Probability. 39. Applied Probability Trust: 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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with finite support |
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with infinite support |
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univariate
continuous- discrete |
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(joint)
- Discrete:
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