Smoothed Estimation of a 3-State Semi-Markov Reliability Model

In this paper, we obtain results involving the estimation of the characteristics of functioning of a reliability system modeled by a homogeneous semi-Markov process (SMP). We use smoothing techniques for estimating the semi-Markov kernel matrix of a finite state homogeneousSMP. In particular, we consider a system with two up or functioning states, and one down or failed state. We assume that transitions between the two up states are allowed, and that the elements of the state space are ordered according to the level of performance of the system that they are representing. We obtain an estimate of the semi-Markov matrix by estimating the transition probability matrix of the embedded Markov chain, and the distribution functions of the transition times between states. The uniformly strong consistency of the estimator is explored. We also construct bootstrap confidence intervals for the elements of the semi-Markov matrix based on the percentile method. Finally, we propose a simplified method to obtain an estimation of the reliability function. The methods presented are easily generalized to more complex situations.