On multi-step MLE-process for Markov sequences (Q310050)

From the text: ``The problem of the construction of the estimator-process of the unknown finite-dimensional parameter in the case of the observations of nonlinear autoregressive process is considered.'' It is supposed that the noise distribution is completely known and the process is stationary with a unique invariant distribution. ``The estimation is done in two or three steps.'' In the two-step procedure the series \(X_0,\dots,X_n\) is split into two parts. The first learning interval \([0,N]\) is used to get a preliminary estimator, e.g. a MLE or a Bayes estimator or a moment estimator. Using this estimator, an iteratively defined MLE process based on the normalized score function is created. In case \(N=n^{\delta}\), \(1/2<\delta < 1\), it is proved that the MLE process is asymptotically normal and efficient in \((0,1]\). In case \(N=n^{\delta}\), \(1/4 \delta \leq 1/2\), the authors suggest a three-step procedure. As the preliminary estimator may not have the right rate of convergence, the second preliminary estimator process is created and then in the third step an asymptotically efficient estimator process is obtained. The procedures are illustrated by two examples. In the first example a two-step procedure is applied with a preliminary estimator obtained by the maximum likelihood method. In the second example a three-step procedure is used with a preliminary moment estimator based on \(N=n^{3/8}\) observations.