A closure characterisation of phase-type distributions

1992 ◽  
Vol 29 (01) ◽  
pp. 92-103 ◽  
Author(s):  
Robert S. Maier ◽  
Colm Art O'Cinneide
Keyword(s):  
Analogous Result ◽  
Finite Automata ◽  
Automata Theory ◽  
Phase Type Distribution ◽  
Exponential Distributions ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Discrete Phase ◽  
Rational Sequence ◽  
Family Of Distributions

We characterise the classes of continuous and discrete phase-type distributions in the following way. They are known to be closed under convolutions, mixtures, and the unary ‘geometric mixture' operation. We show that the continuous class is the smallest family of distributions that is closed under these operations and contains all exponential distributions and the point mass at zero. An analogous result holds for the discrete class. We also show that discrete phase-type distributions can be regarded as ℝ+-rational sequences, in the sense of automata theory. This allows us to view our characterisation of them as a corollary of the Kleene–Schützenberger theorem on the behavior of finite automata. We prove moreover that any summable ℝ+-rational sequence is proportional to a discrete phase-type distribution.

A closure characterisation of phase-type distributions

1992 ◽  
Vol 29 (1) ◽  
pp. 92-103 ◽  
Author(s):  
Robert S. Maier ◽  
Colm Art O'Cinneide
Keyword(s):  
Analogous Result ◽  
Finite Automata ◽  
Automata Theory ◽  
Phase Type Distribution ◽  
Exponential Distributions ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Discrete Phase ◽  
Rational Sequence ◽  
Family Of Distributions

We characterise the classes of continuous and discrete phase-type distributions in the following way. They are known to be closed under convolutions, mixtures, and the unary ‘geometric mixture' operation. We show that the continuous class is the smallest family of distributions that is closed under these operations and contains all exponential distributions and the point mass at zero. An analogous result holds for the discrete class.We also show that discrete phase-type distributions can be regarded as ℝ+-rational sequences, in the sense of automata theory. This allows us to view our characterisation of them as a corollary of the Kleene–Schützenberger theorem on the behavior of finite automata. We prove moreover that any summable ℝ+-rational sequence is proportional to a discrete phase-type distribution.

Presentation of phase-type distributions as proper mixtures

1985 ◽  
Vol 22 (01) ◽  
pp. 247-250 ◽  
Author(s):  
David Assaf ◽  
Naftali A. Langberg
Keyword(s):  
Phase Type Distribution ◽  
Type Distribution ◽  
Phase Type ◽  
Phase Type Distributions

It is shown that any phase-type distribution can be represented as a proper mixture of two distinct phase-type distributions. Using different terms, it is shown that the class of phase-type distributions does not include any extreme ones. A similar result holds for the subclass of upper-triangular phase-type distributions.

scholarly journals Preservation of phase-type distributions under Poisson shock models

1992 ◽  
Vol 24 (01) ◽  
pp. 223-225 ◽  
Author(s):  
M. Manoharan ◽  
Harshinder Singh ◽  
Neeraj Misra
Keyword(s):  
Continuous Phase ◽  
Finite Mixture ◽  
Life Distribution ◽  
Mean Values ◽  
Shock Models ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Discrete Phase ◽  
The Mean ◽  
The Relationship

In this paper, we consider the life distribution H(t) of a device subject to shocks governed by a finite mixture of homogeneous Poisson processes. It will be shown that if (pk ), the probabilities that the device fails on the kth shock, has a discrete phase-type (DPH) distribution, then H(t) is continuous phase-type (CPH). The relationship between the mean values of (pk ) and H(t) is established.

Phase-type distributions and invariant polytopes

1991 ◽  
Vol 23 (3) ◽  
pp. 515-535 ◽  
Author(s):  
Colm Art O'Cinneide
Keyword(s):  
Lower Bounds ◽  
Stieltjes Transform ◽  
Phase Type Distribution ◽  
Type Distribution ◽  
Natural Approach ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Real Poles

The notion of an invariant polytope played a central role in the proof of the characterization of phase-type distributions. The purpose of this paper is to develop invariant polytope techniques further. We derive lower bounds on the number of states needed to represent a phase-type distribution based on poles of its Laplace–Stieltjes transform. We prove that every phase-type distribution whose transform has only real poles has a bidiagonal representation. We close with three short applications of the invariant polytope idea. Taken together, the results of this paper show that invariant polytopes provide a natural approach to many questions about phase-type distributions.

scholarly journals A matrix-analytic approach to the N-player ruin problem

2006 ◽  
Vol 43 (03) ◽  
pp. 755-766 ◽  
Author(s):  
Yvik C. Swan ◽  
F. Thomas Bruss
Keyword(s):  
Computational Procedure ◽  
Phase Type Distribution ◽  
Ruin Time ◽  
Absorbing Markov Chain ◽  
Folding Algorithm ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Set Up ◽  
Different Levels ◽  
Ruin Problem

ConsiderNplayers, respectively owningx1,x2, …,xNmonetary units, who play a sequence of games, winning from and losing to each other integer amounts according to fixed rules. The sequence stops as soon as (at least) one player is ruined. We are interested in the ruin process of theseNplayers, i.e. in the probability that a given player is ruined first, and also in the expected ruin time. This problem is called theN-player ruin problem. In this paper, the problem is set up as a multivariate absorbing Markov chain with an absorbing state corresponding to the ruin of each player. This is then discussed in the context of phase-type distributions where each phase is represented by a vector of sizeNand the distribution has as many absorbing points as there are ruin events. We use this modified phase-type distribution to obtain an explicit solution to theN-player problem. We define a partition of the set of transient states into different levels, and on it give an extension of the folding algorithm (see Ye and Li (1994)). This provides an efficient computational procedure for calculating some of the key measures.

scholarly journals Conditional tail expectations for multivariate phase-type distributions

2005 ◽  
Vol 42 (03) ◽  
pp. 810-825 ◽  
Author(s):  
Jun Cai ◽  
Haijun Li
Keyword(s):  
Extreme Values ◽  
Threshold Value ◽  
Phase Type Distribution ◽  
Distributional Properties ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Dependent Risks ◽  
Explicit Formulae ◽  
Structural Insight ◽  
Conditional Tail Expectation

The conditional tail expectation in risk analysis describes the expected amount of risk that can be experienced given that a potential risk exceeds a threshold value, and provides an important measure of right-tail risk. In this paper, we study the convolution and extreme values of dependent risks that follow a multivariate phase-type distribution, and derive explicit formulae for several conditional tail expectations of the convolution and extreme values for such dependent risks. Utilizing the underlying Markovian property of these distributions, our method not only provides structural insight, but also yields some new distributional properties of multivariate phase-type distributions.

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