Hypoexponential distribution

Hypoexponential
Parameters	$\lambda _{{1}},\dots ,\lambda _{{k}}>0\,$ rates (real)
Support	$x\in [0;\infty )\!$
PDF	Expressed as a phase-type distribution $-{\boldsymbol {\alpha }}e^{{x\Theta }}\Theta {\boldsymbol {1}}$ Has no other simple form; see article for details
CDF	Expressed as a phase-type distribution $1-{\boldsymbol {\alpha }}e^{{x\Theta }}{\boldsymbol {1}}$
Mean	$\sum _{{i=1}}^{{k}}1/\lambda _{{i}}\,$
Median	$\ln(2)\sum _{{i=1}}^{{k}}1/\lambda _{{i}}$
Mode	$(k-1)/\lambda$ if $\lambda _{{k}}=\lambda$ , for all k
Variance	$\sum _{{i=1}}^{{k}}1/\lambda _{{i}}^{2}$
Skewness	$2(\sum _{{i=1}}^{{k}}1/\lambda _{{i}}^{3})/(\sum _{{i=1}}^{{k}}1/\lambda _{{i}}^{2})^{{3/2}}$
Ex. kurtosis	no simple closed form
MGF	${\boldsymbol {\alpha }}(tI-\Theta )^{{-1}}\Theta {\mathbf {1}}$
CF	${\boldsymbol {\alpha }}(itI-\Theta )^{{-1}}\Theta {\mathbf {1}}$

In probability theory the hypoexponential distribution or the generalized Erlang distribution is a continuous distribution, that has found use in the same fields as the Erlang distribution, such as queueing theory, teletraffic engineering and more generally in stochastic processes. It is called the hypoexponetial distribution as it has a coefficient of variation less than one, compared to the hyper-exponential distribution which has coefficient of variation greater than one and the exponential distribution which has coefficient of variation of one.

Overview

The Erlang distribution is a series of k exponential distributions all with rate $\lambda$ . The hypoexponential is a series of k exponential distributions each with their own rate $\lambda_{i}$ , the rate of the $i^{th}$ exponential distribution. If we have k independently distributed exponential random variables ${\boldsymbol {X}}_{{i}}$ , then the random variable,

{\boldsymbol {X}}=\sum _{{i=1}}^{{k}}{\boldsymbol {X}}_{{i}}

is hypoexponentially distributed. The hypoexponential has a minimum coefficient of variation of $1/k$ .

Relation to the phase-type distribution

As a result of the definition it is easier to consider this distribution as a special case of the phase-type distribution. The phase-type distribution is the time to absorption of a finite state Markov process. If we have a k+1 state process, where the first k states are transient and the state k+1 is an absorbing state, then the distribution of time from the start of the process until the absorbing state is reached is phase-type distributed. This becomes the hypoexponential if we start in the first 1 and move skip-free from state i to i+1 with rate $\lambda_{i}$ until state k transitions with rate $\lambda _{k}$ to the absorbing state k+1. This can be written in the form of a subgenerator matrix,

\left[{\begin{matrix}-\lambda _{{1}}&\lambda _{{1}}&0&\dots &0&0\\0&-\lambda _{{2}}&\lambda _{{2}}&\ddots &0&0\\\vdots &\ddots &\ddots &\ddots &\ddots &\vdots \\0&0&\ddots &-\lambda _{{k-2}}&\lambda _{{k-2}}&0\\0&0&\dots &0&-\lambda _{{k-1}}&\lambda _{{k-1}}\\0&0&\dots &0&0&-\lambda _{{k}}\end{matrix}}\right]\;.

For simplicity denote the above matrix $\Theta \equiv \Theta (\lambda _{{1}},\dots ,\lambda _{{k}})$ . If the probability of starting in each of the k states is

{\boldsymbol {\alpha }}=(1,0,\dots ,0)

then $Hypo(\lambda _{{1}},\dots ,\lambda _{{k}})=PH({\boldsymbol {\alpha }},\Theta ).$

Two parameter case

Where the distribution has two parameters ( $\lambda _{1}\neq \lambda _{2}$ ) the explicit forms of the probability functions and the associated statistics are^[1]

CDF: $F(x)=1-{\frac {\lambda _{2}}{\lambda _{2}-\lambda _{1}}}e^{{-\lambda _{1}x}}+{\frac {\lambda _{1}}{\lambda _{2}-\lambda _{1}}}e^{{-\lambda _{2}x}}$

PDF: $f(x)={\frac {\lambda _{1}\lambda _{2}}{\lambda _{1}-\lambda _{2}}}(e^{{-x\lambda _{2}}}-e^{{-x\lambda _{1}}})$

Mean: ${\frac {1}{\lambda _{1}}}+{\frac {1}{\lambda _{2}}}$

Variance: ${\frac {1}{\lambda _{1}^{2}}}+{\frac {1}{\lambda _{2}^{2}}}$

Coefficient of variation: ${\frac {{\sqrt {\lambda _{1}^{2}+\lambda _{2}^{2}}}}{\lambda _{1}+\lambda _{2}}}$

The coefficient of variation is always < 1.

Given the sample mean ( ${\bar {x}}$ ) and sample coefficient of variation ( $c$ ) the parameters $\lambda _{1}$ and $\lambda _{2}$ can be estimated:

$\lambda _{1}={\frac {2}{{\bar {x}}}}\left[1+{\sqrt {1+2(c^{2}-1)}}\right]^{{-1}}$

$\lambda _{2}={\frac {2}{{\bar {x}}}}\left[1-{\sqrt {1+2(c^{2}-1)}}\right]^{{-1}}$

Characterization

A random variable ${\boldsymbol {X}}\sim Hypo(\lambda _{{1}},\dots ,\lambda _{{k}})$ has cumulative distribution function given by,

F(x)=1-{\boldsymbol {\alpha }}e^{{x\Theta }}{\boldsymbol {1}}

and density function,

f(x)=-{\boldsymbol {\alpha }}e^{{x\Theta }}\Theta {\boldsymbol {1}}\;,

where ${\boldsymbol {1}}$ is a column vector of ones of the size k and $e^{{A}}$ is the matrix exponential of A. When $\lambda _{{i}}\neq \lambda _{{j}}$ for all $i\neq j$ , the density function can be written as

f(x)=\sum _{{i=1}}^{k}\lambda _{i}e^{{-x\lambda _{i}}}\left(\prod _{{j=1,j\neq i}}^{k}{\frac {\lambda _{j}}{\lambda _{j}-\lambda _{i}}}\right)=\sum _{{i=1}}^{k}\ell _{i}(0)\lambda _{i}e^{{-x\lambda _{i}}}

where $\ell _{1}(x),\dots ,\ell _{k}(x)$ are the Lagrange basis polynomials associated with the points $\lambda _{1},\dots ,\lambda _{k}$ .

The distribution has Laplace transform of

{\mathcal {L}}\{f(x)\}=-{\boldsymbol {\alpha }}(sI-\Theta )^{{-1}}\Theta {\boldsymbol {1}}

Which can be used to find moments,

E[X^{{n}}]=(-1)^{{n}}n!{\boldsymbol {\alpha }}\Theta ^{{-n}}{\boldsymbol {1}}\;.

General case

In the general case where there are $a$ distinct sums of exponential distributions with rates $\lambda _{1},\lambda _{2},\cdots ,\lambda _{a}$ and a number of terms in each sum equals to $r_{1},r_{2},\cdots ,r_{a}$ respectively. The cumulative distribution function for $t\geq 0$ is given by

F(t)=1-\left(\prod _{{j=1}}^{a}\lambda _{j}^{{r_{j}}}\right)\sum _{{k=1}}^{a}\sum _{{l=1}}^{{r_{k}}}{\frac {\Psi _{{k,l}}(-\lambda _{k})t^{{r_{k}-l}}\exp(-\lambda _{k}t)}{(r_{k}-l)!(l-1)!}},

with

\Psi _{{k,l}}(x)=-{\frac {\partial ^{{l-1}}}{\partial x^{{l-1}}}}\left(\prod _{{j=0,j\neq k}}^{a}\left(\lambda _{j}+x\right)^{{-r_{j}}}\right).

with the additional convention $\lambda _{0}=0,r_{0}=1$ .

Uses

This distribution has been used in population genetics^[2] and queuing theory^[3]^[4]

References

↑ Bolch, Gunter; Greiner, Stefan; de Meer, Hermann; Trivedi, Kishor Shridharbhai (2006). "Chapter 1. Introduction". Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications (2nd ed.). Wiley-Blackwell. doi:10.1002/0471200581.ch1. ISBN 978-0-471-56525-3.
↑ Strimmer K, Pybus OG (2001) "Exploring the demographic history of DNA sequences using the generalized skyline plot", Mol Biol Evol 18(12):2298-305
↑ http://www.few.vu.nl/en/Images/stageverslag-calinescu_tcm39-105827.pdf
↑ Bekker R, Koeleman PM (2011) "Scheduling admissions and reducing variability in bed demand". Health Care Manag Sci, 14(3):237-249

M. F. Neuts. (1981) Matrix-Geometric Solutions in Stochastic Models: an Algorthmic Approach, Chapter 2: Probability Distributions of Phase Type; Dover Publications Inc.
G. Latouche, V. Ramaswami. (1999) Introduction to Matrix Analytic Methods in Stochastic Modelling, 1st edition. Chapter 2: PH Distributions; ASA SIAM,
Colm A. O'Cinneide (1999). Phase-type distribution: open problems and a few properties, Communication in Statistic - Stochastic Models, 15(4), 731–757.
L. Leemis and J. McQueston (2008). Univariate distribution relationships, The American Statistician, 62(1), 45—53.
S. Ross. (2007) Introduction to Probability Models, 9th edition, New York: Academic Press
S.V. Amari and R.B. Misra (1997) Closed-form expressions for distribution of sum of exponential random variables,IEEE Trans. Reliab. 46, 519–522

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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