Matrix-exponential distribution

{{Short description|Absolutely continuous distribution with rational Laplace–Stieltjes transform}}

{{Probability distribution

| name = Matrix-exponential

| type = continuous

| parameters = α, T, s

| support = {{nowrap|x ∈ [0, ∞)}}

| pdf = {{nowrap|α e^{x T}s}}

| cdf = {{nowrap|1 + αe^xTT⁻¹s}}

| mean =

| median =

| mode =

| variance =

| skewness =

| kurtosis =

| entropy =

| mgf =

| char =

| rate =

}}

In probability theory, the matrix-exponential distribution is an absolutely continuous distribution with rational Laplace–Stieltjes transform.{{Cite book | doi = 10.1002/0471667196.ess1092.pub2| chapter = Matrix-Exponential Distributions| title = Encyclopedia of Statistical Sciences| year = 2006| last1 = Asmussen | first1 = S. R. | last2 = o’Cinneide | first2 = C. A. | isbn = 0471667196}} They were first introduced by David Cox in 1955 as distributions with rational Laplace–Stieltjes transforms.{{Cite journal | doi = 10.1080/15326340802232186| title = Characterization of Matrix-Exponential Distributions| journal = Stochastic Models| volume = 24| issue = 3| pages = 339| year = 2008| last1 = Bean | first1 = N. G. | last2 = Fackrell | first2 = M. | last3 = Taylor | first3 = P. }}

The probability density function is $f(x)\; =\; \backslash mathbf\{\backslash alpha\}\; e^\{x\backslash ,T\}\; \backslash mathbf\{s\}\; \backslash text\{\; for\; \}x\backslash ge\; 0$ (and 0 when x < 0), and the cumulative distribution function is $F(t)\; =\; 1\; -\; \backslash alpha\; e^\{\backslash textbf\{A\}t\}\; \backslash textbf\{1\}${{Cite web |title=Tools for Phase-Type Distributions (butools.ph) — butools 2.0 documentation |url=http://webspn.hit.bme.hu/~telek/tools/butools/doc/ph.html |access-date=2022-04-16 |website=webspn.hit.bme.hu}} where 1 is a vector of 1s and

: $$

\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}

There are no restrictions on the parameters α, T, s other than that they correspond to a probability distribution.{{Cite journal | doi = 10.1239/aap/1175266478| title = On matrix exponential distributions| journal = Advances in Applied Probability| volume = 39| pages = 271–292| year = 2007| last1 = He | first1 = Q. M. | last2 = Zhang | first2 = H. | publisher = Applied Probability Trust| doi-access = free}} 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.