Minimax sequential estimation plans for exponential-type processes

The study of the theory of sequential estimation for continuous-time stochastic processes was initiated by Dvoretzky, Kiefer and Wolfowitz (1953). They proved that for the Poisson, negative-binomial, gamma and Wiener processes the minimax sequential plan reduces to a fixed-time plan if a weighted quadratic loss function is used. Their results were generalized by Magiera (1977) to an exponential class of processes with stationary independent increments and by Trybula (1985) in case of the n-dimensional processes. It turns out, however, that for a certain class of processes there exist truly sequential minimax plans. It was showed by Trybula (1985) that for the Poisson process an inverse plan is minimax. Wilczynski (1985) has obtained an analogous result for the multinomial process. In this paper we consider a class of exponential-type processes and for this class we prove that, with a properly weighted quadratic loss function, the inverse plans are minimax in the class of all sequential plans having finite risk. Examples illustrating the derived result are also presented.