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The estimation problem for a non-linear stochastic model with control is studied. We suppose that the drift of the model depends on the value of an unknown parameter and a control which is a function of the preceding observations. The goal is to construct strategies including a control rule, a stopping time and an estimator of the parameter that are optimal in the minimax sense. The exhaustive solution of this problem is given for the particular case when the drift is a linear function of the parameter. For the general case the asymptotic approach is used when the average observation time tends to infinity. Lower bounds of asymptotic minimax risk are stated for arbitrary sequential strategies. Then an explicit construction of asymptotically optimal strategies is given. These are so called “two-stage strategies”. At the first stage one can choose any non-degenerate plan: the purpose of the first stage is only to construct a preliminary consistent estimator. A design for the second stage (taking most of the...

We consider the problems of minimax estimation in an asymptotic setting for quite general statistical models. In particular, we do not require that the well-known LAN condition holds. We show that general statistical experiments which correspond to a random number of dependent observations can be approximated by experiments of a special form called -models. Then we obtain lower bounds for minimax estimation risk for -models. This allows us to obtain lower bounds for asymptotic minimax risk for a wide class of statistical models