Matrix-exponential distribution

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) where

{\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.^[3] 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]

{\mathbb 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.^[4]

Software

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

References

1 2 Asmussen, S. R.; o’Cinneide, C. A. (2006). "Matrix-Exponential Distributions". Encyclopedia of Statistical Sciences. doi:10.1002/0471667196.ess1092.pub2. ISBN 0471667196.
1 2 3 Bean, N. G.; Fackrell, M.; Taylor, P. (2008). "Characterization of Matrix-Exponential Distributions". Stochastic Models. 24 (3): 339. doi:10.1080/15326340802232186.
↑ 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.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Matrix-exponential distribution

Moments

Fitting

Software

See also

References