hypoexponential distribution

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

:$$

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

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

As a result of the definition it is easier to consider this distribution as a special case of the phase-type distribution.{{cite journal |last1=Legros |first1=Benjamin |last2=Jouini |first2=Oualid |date=2015 |title=A linear algebraic approach for the computation of sums of Erlang random variables |journal=Applied Mathematical Modelling |volume=39 |issue=16 |pages=4971–4977 |doi=10.1016/j.apm.2015.04.013 |doi-access=free}} 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 $\backslash lambda\_\{i\}$ until state k transitions with rate $\backslash lambda\_\{k\}$ 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 $\backslash Theta\backslash equiv\backslash Theta(\backslash lambda\_\{1\},\backslash dots,\backslash lambda\_\{k\})$. If the probability of starting in each of the k states is

:$$

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

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

Where the distribution has two parameters ($\backslash lambda\_1\; \backslash neq\; \backslash lambda\_2$) the explicit forms of the probability functions and the associated statistics are:{{cite book |last1=Bolch |first1=Gunter |last2=Greiner |first2=Stefan |last3=de Meer |first3=Hermann |last4=Trivedi |first4=Kishor S. |author4-link=Kishor S. Trivedi |date=2006 |title=Queuing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications |edition=2nd |publisher=Wiley |pages=24–25 |doi=10.1002/0471791571 |isbn=978-0-471-79157-7}}

CDF: $F(x)\; =\; 1\; -\; \backslash frac\{\backslash lambda\_2\}\{\backslash lambda\_2-\backslash lambda\_1\}e^\{-\backslash lambda\_1x\}\; -\; \backslash frac\{\backslash lambda\_1\}\{\backslash lambda\_1-\backslash lambda\_2\}e^\{-\backslash lambda\_2x\}$

PDF: $f(x)\; =\; \backslash frac\{\backslash lambda\_1\backslash lambda\_2\}\{\backslash lambda\_1-\backslash lambda\_2\}(\; e^\{-x\; \backslash lambda\_2\}\; -\; e^\{-x\; \backslash lambda\_1\}\; )$

Mean: $\backslash frac\{1\}\{\backslash lambda\_1\}+\backslash frac\{1\}\{\backslash lambda\_2\}$

Variance: $\backslash frac\{1\}\{\backslash lambda\_1^2\}+\backslash frac\{1\}\{\backslash lambda\_2^2\}$

Coefficient of variation: $\backslash frac\{\backslash sqrt\{\backslash lambda\_1^2\; +\; \backslash lambda\_2^2\}\}\{\; \backslash lambda\_1\; +\; \backslash lambda\_2\; \}$

The coefficient of variation is always less than 1.

Given the sample mean ($\backslash bar\{x\}$) and sample coefficient of variation ($c$), the parameters $\backslash lambda\_1$ and $\backslash lambda\_2$ can be estimated as follows:

$\backslash lambda\_1=\; \backslash frac\{\; 2\}\{\; \backslash bar\{x\}\; \}\; \backslash left[\; 1\; +\; \backslash sqrt\{\; 1\; +\; 2\; (\; c^2\; -\; 1\; )\; \}\; \backslash right]^\{-1\}$

$\backslash lambda\_2\; =\; \backslash frac\{\; 2\; \}\{\; \backslash bar\{x\}\; \}\; \backslash left[\; 1\; -\; \backslash sqrt\{\; 1\; +\; 2\; (\; c^2\; -\; 1\; )\; \}\; \backslash right]^\{-1\}$

These estimators can be derived from the methods of moments by setting $\backslash frac\{1\}\{\backslash lambda\_1\}+\backslash frac\{1\}\{\backslash lambda\_2\}=\backslash bar\; x$

and $$

\frac{\sqrt{\lambda_1^2+\lambda_2^2}}{\lambda_1+\lambda_2}=c

.

The resulting parameters $\backslash lambda\_1$ and $\backslash lambda\_2$ are real values if $c^2\backslash in[0.5,1]$.

A random variable $\backslash boldsymbol\{X\}\backslash sim\; Hypo(\backslash lambda\_\{1\},\backslash dots,\backslash lambda\_\{k\})$ 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 $\backslash boldsymbol\{1\}$ is a column vector of ones of the size k and $e^\{A\}$ is the matrix exponential of A. When $\backslash lambda\_\{i\}\; \backslash ne\; \backslash lambda\_\{j\}$ for all $i\; \backslash ne\; j$, the density function can be written as

:$$

f(x) = \sum_{i=1}^k \lambda_i e^{-x \lambda_i} \left(\prod_{j=1, j \ne i}^k \frac{\lambda_j}{\lambda_j - \lambda_i}\right) = \sum_{i=1}^k \ell_i(0) \lambda_i e^{-x \lambda_i}

where $\backslash ell\_1(x),\; \backslash dots,\; \backslash ell\_k(x)$ are the Lagrange basis polynomials associated with the points $\backslash lambda\_1,\backslash dots,\backslash lambda\_k$.

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}\; .

In the general case

where there are $a$ distinct sums of exponential distributions

with rates $\backslash lambda\_1,\backslash lambda\_2,\backslash cdots,\backslash lambda\_a$ and a number of terms in each

sum equals to $r\_1,r\_2,\backslash cdots,r\_a$ respectively. The cumulative

distribution function for $t\backslash geq0$ 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

:$\backslash Psi\_\{k,l\}(x)$

= -\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 $\backslash lambda\_0\; =\; 0,\; r\_0\; =\; 1$.{{cite journal |last1=Amari |first1=Suprasad V. |last2=Misra |first2=Ravindra B. |date=1997 |title=Closed-form expressions for distribution of sum of exponential random variables |journal=IEEE Transactions on Reliability |volume=46 |issue=4 |pages=519–522 |doi=10.1109/24.693785}}

This distribution has been used in population genetics,{{cite journal |last1=Strimmer |first1=Korbinian |last2=Pybus |first2=Oliver G. |date=2001 |title=Exploring the demographic history of DNA sequences using the generalized skyline plot |journal=Molecular Biology and Evolution |volume=18 |issue=12 |pages=2298–2305 |doi=10.1093/oxfordjournals.molbev.a003776 |doi-access=free |pmid= 11719579}} cell biology,{{cite journal |last1=Yates |first1=Christian A. |last2=Ford |first2=Matthew J. |last3=Mort |first3=Richard L. |date=2017 |title=A multi-stage representation of cell proliferation as a Markov process |journal=Bulletin of Mathematical Biology |volume=79 |issue=12 |pages=2905–2928 |arxiv=1705.09718 |doi=10.1007/s11538-017-0356-4 |doi-access=free |pmc=5709504 |pmid=29030804}}{{cite journal |last1=Gavagnin |first1=Enrico |last2=Ford |first2=Matthew J. |last3=Mort |first3=Richard L. |last4=Rogers |first4=Tim |last5=Yates |first5=Christian A. |date=2019 |title=The invasion speed of cell migration models with realistic cell cycle time distributions |journal=Journal of Theoretical Biology |volume=481 |pages=91–99 |arxiv=1806.03140 |doi=10.1016/j.jtbi.2018.09.010 |pmid=30219568}} and queuing theory.{{cite web |url=http://www.few.vu.nl/en/Images/stageverslag-calinescu_tcm39-105827.pdf |title=Forecasting and capacity planning for ambulance services |last=Călinescu |first=Malenia |date=August 2009 |website=Faculty of Sciences |publisher=Vrije Universiteit Amsterdam |archive-url=https://web.archive.org/web/20100215173841/http://www.few.vu.nl/en/Images/stageverslag-calinescu_tcm39-105827.pdf |archive-date=15 February 2010}}{{cite journal |last1=Bekker |first1=René |last2=Koeleman |first2=Paulien M. |date=2011 |title=Scheduling admissions and reducing variability in bed demand |journal=Health Care Management Science |volume=14 |issue=3 |pages=237–249 |doi=10.1007/s10729-011-9163-x |doi-access=free |pmc=3158339 |pmid=21667090}}

• Phase-type distribution
• Coxian distribution

{{reflist}}

• 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

{{ProbDistributions|continuous-semi-infinite}}

{{DEFAULTSORT:Hypoexponential Distribution}}

Category:Continuous distributions

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