A stationary Weibull frailty process

In this paper we have developed two correlated stationary processes Xn and Yn, where P(Xn=Xn+1)>0 and P(Yn=Yn+1)>0. The model has been developed based on two real life data sets. Efficient inference procedures and goodness of fit test have been developed. The motivation of this work came when we were trying to analyse the gold price data {Xn} of the Indian market and the exchange rate data {Yn} between Indian Rupees and US dollars during the same time period. Although, both of them are stationary processes, it is observed that there is a significant amount of time Xn=Xn+1 or Yn=Yn+1, and they cannot be ignored. Moreover, Xn and Yn are dependent, and if we want to analyse the data during the same period, they should be analysed together. In this paper first we have proposed a new stationary process based on Marshall–Olkin bivariate exponential distribution type construction combining with frailty, so that P(Xn=Xn+1)>0. Different properties of the proposed bivariate stationary process have been established. The maximum likelihood estimators of the unknown parameters have been obtained. Goodness of fit test of the proposed model has been proposed. We have performed the analysis of gold price data and the exchange rate data using the proposed model and the results are quite satisfactory. Then we have extended the process to the bivariate case where P(Xn=Xn+1)>0, P(Yn=Yn+1)>0 and Xn, Yn are dependent. We have demonstrated how it can be used in practice.