Matrix-exponential distribution

Absolutely continuous distribution with rational Laplace–Stieltjes transform

Matrix-exponential
Parameters	α, T, s
Support	x ∈ [0, ∞)
PDF	α e^{x T}s
CDF	1 + αe^xTT⁻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 $f(x)=\mathbf {\alpha } e^{x\,T}\mathbf {s} {\text{ for }}x\geq 0$ (and 0 when x < 0), and the cumulative distribution function is $F(t)=1-\alpha e^{{\textbf {A}}t}{\textbf {1}}$ ^[3] where 1 is a vector of 1s and

{\begin{aligned}\alpha &\in \mathbb {R} ^{1\times n},\\T&\in \mathbb {R} ^{n\times n},\\s&\in \mathbb {R} ^{n\times 1}.\end{aligned}}

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]

\operatorname {E} (X^{k})=(-1)^{k+1}k!\mathbf {\alpha } T^{-(k+1)}\mathbf {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.

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.

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric negative Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic mixed Poisson negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous
univariate

supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi chi-squared noncentral inverse scaled Dagum Davis Erlang hyper exponential hyperexponential hypoexponential logarithmic F noncentral folded normal Fréchet gamma generalized inverse gamma/Gompertz Gompertz shifted half-logistic half-normal Hotelling's T-squared inverse Gaussian generalized Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal log-t Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice truncated normal type-2 Gumbel Weibull discrete Wilks's lambda
supported on the whole real line	Cauchy exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t Tracy–Widom variance-gamma Voigt
with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda