Hypoexponential distribution

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 X_{i}, then the random variable,

: X=\sum^{k}{i=1}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:

CDF: F(x) = 1 - \frac{\lambda_2}{\lambda_2-\lambda_1}e^{-\lambda_1x} - \frac{\lambda_1}{\lambda_1-\lambda_2}e^{-\lambda_2x}

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 less than 1.

Given the sample mean (\bar{x}) and sample coefficient of variation (c), the parameters \lambda_1 and \lambda_2 can be estimated as follows:

\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}

These estimators can be derived from the methods of moments by setting \frac{1}{\lambda_1}+\frac{1}{\lambda_2}=\bar x

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

.

The resulting parameters \lambda_1 and \lambda_2 are real values if c^2\in[0.5,1].

Characterization

A random variable 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} \ne \lambda_{j} for all i \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 \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\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

:\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, cell biology, and queuing theory.

References

References

1. (2012). "HypoexponentialDistribution". Wolfram.
2. (2015). "A linear algebraic approach for the computation of sums of Erlang random variables". Applied Mathematical Modelling.
3. (2006). "Queuing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications". Wiley.
4. (1997). "Closed-form expressions for distribution of sum of exponential random variables". IEEE Transactions on Reliability.
5. (2001). "Exploring the demographic history of DNA sequences using the generalized skyline plot". [[Molecular Biology and Evolution]].
6. (2017). "A multi-stage representation of cell proliferation as a Markov process". Bulletin of Mathematical Biology.
7. (2019). "The invasion speed of cell migration models with realistic cell cycle time distributions". [[Journal of Theoretical Biology]].
8. Călinescu, Malenia. (August 2009). "Forecasting and capacity planning for ambulance services". [[Vrije Universiteit Amsterdam]].
9. (2011). "Scheduling admissions and reducing variability in bed demand". Health Care Management Science.