Bayesian Nonparametric Estimation of Intensity Functions Using Markov Chain Monte Carlo Method

Let N(t) be a counting process indicating the failures (and instantaneous repairs) of a repairable system up to time t. We model the probabilistic behaviour of the process by a nonhomogeneous Poisson process (NHHP) with intensity function λ(t). In the absence of any grounds in favour of a specific parametric form for λ(t) we assume that it is constant between failure epochs and at these epochs receives a jump (the damage-atfrulure- model). Again we don't know the heights of these jumps. Therefore we assume a heirarchical Bayes formulation. The heights of the jumps are assumed to have constrained conditional gamma distributions. The hyperparameters of the gamma distribution are assumed to have known uniform distributions. The exact Bayes solutions are analytically intractable. Hence we provide a scheme of simulation via the Metropolis-Hastings algorithm to estimate the features of the posterior distribution. The methodology is illustrated by applying it to a well known data set.

AMS (2000) Subject Classification: Primary 62F15, 62G05; Secondary 65C05.