Minimax sequential procedures for Markov renewal processes

The paper deals with the problem of finding minimax sequential procedures for Markov renewal processes. To find the optimal sequential procedures in some statistical models for these processes, a tool of exponential families is used. A minimax theorem is presented which is useful in deriving the optimal sequential estimation procedures for a multiparameter exponential model for stochastic processes, when a weighted squared error loss is used and the cost of observing the process is taken into account. Using this tool, a class of minimax sequential procedures is derived for estimating the ratios between transition probabilities of the embedded Markov chain and the mean value parameter of the additive part of the Markov renewal processes considered.