Vlad Stefan Barbu

Semi-Markov Chains and Hidden Semi-Markov Models toward Applications: Their Use in Reliability and DNA Analysis

This book is concerned with the estimation of discrete-time semi-Markov and hidden semi-Markov processes. Semi-Markov processes are much more general and better adapted to applications than the Markov ones because sojourn times in any state can be arbitrarily distributed, as opposed to the geometrically distributed sojourn time in the Markov case. Another unique feature of the book is the use of discrete time, especially useful in some specific applications where the time scale is intrinsically discrete. The models presented in the book are specifically adapted to reliability studies and DNA analysis. The book is mainly intended for applied probabilists and statisticians interested in semi-Markov chains theory, reliability and DNA analysis, and for theoretical oriented reliability and bioinformatics engineers. It can also serve as a text for a six month research-oriented course at a Master or PhD level. The prerequisites are a background in probability theory and finite state space Markov chains.

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

Abstract In this paper, we define a discrete-time semi-Markov model and propose a computation procedure for solving the corresponding Markov renewal equation, necessary for all our reliability measurements. Then, we compute the reliability and its related measures, and we apply the results to a three-state system.

Empirical estimation for discrete-time semi-Markov processes with applications in reliability

We consider a discrete-time semi-Markov process, with a finite state space. Taking a censored history, we obtain empirical estimators for the semi-Markov kernel, semi-Markov transition function, reliability and availability. We study the strong consistency and the asymptotic normality for each estimator.

Nonparametric Estimation for Failure Rate Functions of Discrete Time semi-Markov Processes

We consider a semi-Markov chain, with a finite state space. Taking a censored history, we obtain empirical estimators for the discrete semi-Markov kernel, renewal function and semi-Markov transition function. We propose estimators for two different failure rate functions: the usual failure rate, BMP-failure rate, defined by [BMP63], and the new introduced failure rate, RG-failure rate, proposed by [RG92]. We study the strong consistency and the asymptotic normality for each estimator and we construct the confidence intervals. We illustrate our results by a numerical example.

Discrete Time Semi-Markov Processes for Reliability and Survival Analysis — A Nonparametric Estimation Approach

In the first part of this paper, after recalling the definition and properties of a discrete time semi-Markov model with a finite state space, we propose a computation procedure for solving the corresponding Markov renewal equation, appearing in survival analysis. This provides us an easily computable formula for the survival function. In the second part, we consider a censored history and we propose empirical estimators for the semi-Markov kernel, Semi-Markov transition matrix and survival or reliability. We study the strong consistency and the asymptotic normality of the estimator of survival function. We apply the results to a three state system.

Reliability of Semi-Markov Systems in Discrete Time: Modeling and Estimation

This chapter presents the reliability of discrete-time semi-Markov systems. After some basic definitions and notation, we obtain explicit forms for reliability indicators. We propose non-parametric estimators for reliability, availability, failure rate, mean hitting times and we study their asymptotic properties. Finally, we present a three state example with detailed calculations and numerical evaluations.

Robust adaptive efficient estimation for semi-Markov nonparametric regression models

We consider the nonparametric robust estimation problem for regression models in continuous time with semi-Markov noises. An adaptive model selection procedure is proposed. Under general moment conditions on the noise distribution a sharp non-asymptotic oracle inequality for the robust risks is obtained and the robust efficiency is shown. It turns out that for semi-Markov models the robust minimax convergence rate may be faster or slower than the classical one.

On Semi-Markov Modelling and Inference for Multi-state Systems

In this work we focus on multi state systems that we model by means of semi-Markov processes. The sojourn times are seen to be independent not identically distributed random variables and assumed to belong to a general class of distributions that includes several popular reliability distributions like the exponential, Weibull, and Pareto. We obtain maximum likelihood estimators of the parameters of interest and for various quantities related to the system under study.

Modeling and Inference for Multi-state Systems

In this work we are focused on multi-state systems modeled by means of a special type of semi-Markov processes. The sojourn times are seen to be independent not necessarily identically distributed random variables and assumed to belong to a general class of distributions closed under extrema that includes, in addition to some discrete distributions, several typical reliability distributions like the exponential, Weibull, and Pareto. A special parametrization is proposed for the parameters describing the system, taking thus into account various types of dependencies of the parameters on the the states of the system. We obtain maximum likelihood estimators of the parameters and plug-in type estimators are furnished for the basic quantities describing the semi-Markov system under study.

Semi-Markov Modelling for Multi-state Systems

Markov processes are widely used in reliability engineering. In this work we focus on multi state systems (MSS) and apply the Semi-Markov methodology for parameter estimation. For this purpose the sojourn times are assumed to be independent not identically distributed (inid) random variables that follow a general class of distributions that includes several popular reliability distributions like the exponential, Weibull, and Pareto.