Discrete-Time Semi-Markov Model for Reliability and Survival Analysis

Barbu, Vlad Stefan; Boussemart, Michel; Limnios, Nikolaos

doi:10.1081/sta-200037923

Cited by 91 publications

(56 citation statements)

References 11 publications

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“…The summation s≥k1 concerns the fact that we do not know what is the letter which appears before the first letter of the word w (which is also the first letter of the run word w 1 ). On the other hand, we know what is the letter which comes after the last letter b 1 of the run word w (1). For the runs w(2), .…”

Section: If W Is Not a Run That Is W = W(1)w(2) W(η)mentioning

confidence: 99%

“…(see [1]). For p * (α, α ) > 0 the Markov chain {Z γ ; γ ∈ N} is irreducible and aperiodic with stationary distribution…”

Section: If W Is Not a Run That Is W = W(1)w(2) W(η)mentioning

confidence: 99%

“…Recently, Barbu et al [1] studied a discrete time semi-Markov process {Z γ ; γ ∈ N} in order to derive some specific reliability measurements. Chryssaphinou et al [2] defining the process {U γ ; γ ∈ N} to be the backward recurrence times of the DTSM process {Z γ ; γ ∈ N} studied the Markov process {(Z γ , U γ ); γ ∈ N}.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the number of word occurrences in a semi-Markov sequence of letters

Karaliopoulou

2009

ESAIM: PS

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Abstract. Let a finite alphabet Ω. We consider a sequence of letters from Ω generated by a discrete time semi-Markov process {Zγ ; γ ∈ N}. We derive the probability of a word occurrence in the sequence. We also obtain results for the mean and variance of the number of overlapping occurrences of a word in a finite discrete time semi-Markov sequence of letters under certain conditions. Mathematics Subject Classification. 60K10, 60K20, 60C05, 60E05.

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Section: If W Is Not a Run That Is W = W(1)w(2) W(η)mentioning

confidence: 99%

“…(see [1]). For p * (α, α ) > 0 the Markov chain {Z γ ; γ ∈ N} is irreducible and aperiodic with stationary distribution…”

Section: If W Is Not a Run That Is W = W(1)w(2) W(η)mentioning

confidence: 99%

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On the number of word occurrences in a semi-Markov sequence of letters

Karaliopoulou

2009

ESAIM: PS

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“…As the states of X n are mainly defined as the state combinations of ζ n and η n , the calculation of the semi-Markov kernel of X n is simplified when ρ (i, j) (k,l) (m) is available, as shown in the following listed formulas: Although these formulas appear to be simpler, both the parameter estimation and competition probability calculations need much more calculation burden than the first case when FDI decision is modeled as a hypothetical Markov chain. Once X n is constructed, calculation of reliability function and MTTF are straightforward using available formulas [9].…”

Section: Definitionmentioning

confidence: 99%

Reliability Monitoring of Fault Tolerant Control Systems with Demonstration on an Aircraft Model

Zhao

Yang

2008

Journal of Control Science and Engineering

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This paper proposes a reliability monitoring scheme for active fault tolerant control systems using a stochastic modeling method. The reliability index is defined based on system dynamical responses and a safety region; the plant and controller are assumed to have a multiple regime model structure, and a semi-Markov model is built for reliability evaluation based on the safety behavior of each regime model estimated by using Monte Carlo simulation. Moreover, the history data of fault detection and isolation decisions is used to update its transition characteristics and reliability model. This method provides an up-to-date reliability index as demonstrated on an aircraft model.

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“…For recent references on theory and applications (e.g. reliability) of discrete SMCs see Barbu and Limnios (2008) and Barbu et al (2004).…”

Section: Introductionmentioning

confidence: 99%

Exact MLE and asymptotic properties for nonparametric semi-Markov models

Trevezas

Limnios

2011

Journal of Nonparametric Statistics

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This article concerns maximum likelihood estimation for discrete time homogeneous nonparametric semi-Markov models with finite state space. In particular, we present the exact maximum likelihood estimator of the semi-Markov kernel which governs the evolution of the semi-Markov chain. We study its asymptotic properties in the following cases: (i) for one observed trajectory, when the length of the observation tends to infinity, and (ii) for parallel observations of independent copies of a semi-Markov chain censored at a fixed time, when the number of copies tends to infinity. In both cases, we obtain strong consistency, asymptotic normality, and asymptotic efficiency for every finite dimensional vector of this estimator. Finally, we obtain explicit forms for the covariance matrices of the asymptotic distributions.

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Discrete-Time Semi-Markov Model for Reliability and Survival Analysis

Cited by 91 publications

References 11 publications

On the number of word occurrences in a semi-Markov sequence of letters

On the number of word occurrences in a semi-Markov sequence of letters

Reliability Monitoring of Fault Tolerant Control Systems with Demonstration on an Aircraft Model

Exact MLE and asymptotic properties for nonparametric semi-Markov models

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