The motivation of this work came when we were trying to analyze gold price data of the Indian market and the exchange rate data between Indian Rupees and US Dollars. It is observed that in both the cases there is a significant amount of time when \(X_n = X_{n+1}\) , hence they cannot be ignored. In this paper we have introduced a very flexible discrete time and continuous state space stationary stochastic process \(\{X_n\}\) , where \(X_n\) has a proportional hazard class of distribution and there is a positive probability that \(X_n = X_{n+1}\) . We have assumed a very flexible piecewise constant hazard function of the base line distribution of the proportional hazard class. Various properties of the proposed class has been obtained. Various dependency properties have been established. Estimating the cut points of the piecewise constant hazard function is an important problem and it has been addressed here. The maximum likelihood estimators (MLEs) of the unknown parameters cannot be obtained in closed form, and we have proposed to use profile likelihood method to compute the estimators. The gold price data set and the exchange rate data set have been analyzed and the results are quite satisfactory.
