Estimation of stress-strength reliability using discrete phase type distribution

AbstractIn this paper, a generalized class of run shock models associated with a bivariate sequence {(Xi, Yi)}i≥1 of correlated random variables is defined and studied. For a system that is subject to shocks of random magnitudes X1, X2, ... over time, let the random variables Y1, Y2, ... denote times between arrivals of successive shocks. The lifetime of the system under this class is defined through a compound random variable T = ∑Nt=1 Yt , where N is a stopping time for the sequence {Xi}i≤1 and represents the number of shocks that causes failure of the system. Another random variable of interest is the maximum shock size up to N, i.e. M = max {Xi, 1≤i≤ N}. Distributions of T and M are investigated when N has a phase-type distribution.

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A closure characterisation of phase-type distributions

Journal of Applied Probability ◽

10.1017/s0021900200106655 ◽

1992 ◽

Vol 29 (01) ◽

pp. 92-103 ◽

Cited By ~ 1

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.

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Time-dependent stress–strength reliability models based on phase type distribution

Computational Statistics ◽

10.1007/s00180-020-00991-3 ◽

2020 ◽

Vol 35 (3) ◽

pp. 1345-1371 ◽

Cited By ~ 1

Author(s):

Joby K. Jose ◽

M. Drisya

Keyword(s):

Time Dependent ◽

Phase Type Distribution ◽

Type Distribution ◽

Reliability Models ◽

Phase Type

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Presentation of phase-type distributions as proper mixtures

Journal of Applied Probability ◽

10.1017/s0021900200029193 ◽

1985 ◽

Vol 22 (01) ◽

pp. 247-250 ◽

Cited By ~ 1

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.

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Ruin Probability for Stochastic Flows of Financial Contract under Phase-Type Distribution

Risks ◽

10.3390/risks8020053 ◽

2020 ◽

Vol 8 (2) ◽

pp. 53

Author(s):

Franck Adékambi ◽

Kokou Essiomle

Keyword(s):

Ruin Probability ◽

Upper And Lower Bounds ◽

Ruin Probabilities ◽

Adjustment Coefficient ◽

Phase Type Distribution ◽

Type Distribution ◽

Andersen Model ◽

Phase Type ◽

Financial Contract ◽

The Impact

This paper examines the impact of the parameters of the distribution of the time at which a bank’s client defaults on their obligated payments, on the Lundberg adjustment coefficient, the upper and lower bounds of the ruin probability. We study the corresponding ruin probability on the assumption of (i) a phase-type distribution for the time at which default occurs and (ii) an embedding of the stochastic cash flow or the reserves of the bank to the Sparre Andersen model. The exact analytical expression for the ruin probability is not tractable under these assumptions, so Cramér Lundberg bounds types are obtained for the ruin probabilities with concomitant explicit equations for the calculation of the adjustment coefficient. To add some numerical flavour to our results, we provide some numerical illustrations.

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