In this paper we consider one discrete time and continuous state space stationary stochastic process \(\{X_n; n \ge 1\}\) , where \(X_n\) ’s are positive valued random variables. Further, the marginal distribution of \(X_n\) belongs to the proportional hazard class of distributions, and there is a positive probability that \(X_n = X_{n+1}\) . We provide different properties of the proposed process. We consider different distributions of \(X_n\) , namely (a) exponential, (b) Weibull and (c) piecewise constant hazard function. All these distributions belong to the proportional hazard class of distributions. We provide the Bayesian inference of the unknown parameters in all these cases under a very flexible set of priors. The Bayes estimators and the associated credible intervals cannot be obtained in closed forms. We propose to use very convenient importance sampling technique to compute the Bayes estimators and the associated credible intervals. We consider one gold price data set of the Indian market and one exchange rate data set between Indian Rupees and US dollars and in both these data sets there is a significant number of n, for which \(X_n = X_{n+1}\) , hence they cannot be ignored. Since, the proposed model is capable of handing this issue, we use the proposed model to analyze both the data sets. The results are quite satisfactory.
