Non-parametric estimation for semi-Markov kernels with application to reliability analysis
✍ Scribed by Ouhbi, B. ;Limnios, N.
- Publisher
- John Wiley and Sons
- Year
- 1996
- Tongue
- English
- Weight
- 408 KB
- Volume
- 12
- Category
- Article
- ISSN
- 8755-0024
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✦ Synopsis
The authors consider an irreducible Markov renewal process (MRP) with a finite number of states. Their aim is to derive estimators of a censored MRP with a finite number of states either in a fixed time T or in the Nth jump. The estimators given here are seen to be of the Kaplan-Meier type. The asymptotic properties these estimators are given. The reliability of a semi-Markov system model is examined numerically by the estimators, and a comparison is made with estimators obtained by Lagakos et al. KEY WORDS Markov renewal process; estimation of the transition function; Kaplan-Meier estimator; reliability 1. INTRODUCTION Markov renewal processes (MRPs) are useful in random phenomena modelling studies: reliability of systems, evolution of populations, medical treatments, etc. (cf. Limnios and Oprisanl), characterized by:
(i) for a subject in state i at the nth jump, the probability that the next visited state is j does not depend on the history before his entry in state i; (ii) for a subject in state i at the nth jump, on condition that the next visited state is j , the distribution of the sojourn time in state i does not depend on the history before his entry in state i.
This model is determined if one can give its semi-Markov kernel and its initial law. Many papers have given estimators of the kernel of the MRP (cf. Moore and F'yke,* Lagakos ef by the maximum likelihood estimation method (MLEM). Let us consider an MRP { (J,, S, ), n 2 0 J defined on a probability complete space with state J , to be in E = { 1,2, . . . , s J (the system state space), sojourn time X, between two jumps to be in R + and S, = XI +. . .+ X,, n 5 1. X,=Oa.s. a n d P [ J , = k ] = p ( k ) P [ J , = k , X, Q x / J , , J,, . . ., Jn-lr XI, X,, . . ., X,-l 1 = Q(J,-I , k, x ) a.s. for all x E [0, +=I and 1 S k s s. Let p ( i , j ) = Q(i,j, =) (=limQ(i,j, t ) as t+=)