Recursive construction of confidence regions

Abstract

Assuming that one-step transition kernel of a discrete time, time-homogenous Markov chain model is parameterized by a parameter $\theta\in\boldsymbol{\Theta}$, we derive a recursive (in time) construction of confidence regions for the unknown parameter of interest, say $\theta^{*}\in\boldsymbol{\Theta}$. It is supposed that the observed data used in the construction of the confidence regions is generated by a Markov chain whose transition kernel corresponds to $\theta^{*}$. The key step in our construction is the derivation of a recursive scheme for an appropriate point estimator of $\theta^{*}$. To achieve this, we start by what we call the base recursive point estimator, using which we design a quasi-asymptotically linear recursive point estimator (a concept introduced in this paper). For the latter estimator we prove its weak consistency and asymptotic normality. The recursive construction of confidence regions is needed not only for the purpose of speeding up the computation of the successive confidence regions, but, primarily, for the ability to apply the dynamic programming principle in the context of robust adaptive stochastic control methodology.