A stationary Weibull process and its applications

In this paper we introduce a discrete-time and continuous state-space Markov stationary process $ \{X_n; n = 1, 2, \ldots \} $ {Xn;n=1,2,…}, where $ X_n $ Xn has a two-parameter Weibull distribution, $ X_n $ Xn's are dependent and there is a positive probability that $ X_n = X_{n+1} $ Xn=Xn+1. The motivation came from the gold price data where there are several instances for which $ X_n = X_{n+1} $ Xn=Xn+1. Hence, the existing methods cannot be used to analyze this data. We derive different properties of the proposed Weibull process. It is observed that the joint cumulative distribution function of $ X_n $ Xn and $ X_{n+1} $ Xn+1 has a very convenient copula structure. Hence, different dependence properties and dependence measures can be obtained. The maximum likelihood estimators cannot be obtained in explicit forms, we have proposed a simple profile likelihood method to compute these estimators. We have used this model to analyze two synthetic data sets and one gold price data set of the Indian market, and it is observed that the proposed model fits quite well with the data set.