Some Aspects of Statistical Inference in a Markovian and Mixing Framework

This paper is a contribution to a special volume in memory of our departed colleague, Sidney Yakowitz, of the University of Arizona, Tucson. The material discussed is taken primarily from existing literature in Markovian and mixing processes. Emphasis is given to the statistical inference aspects of such processes. In the Markovian case, both parametric and nonparametric inferences are considered. In the parametric component, the classical approach is used, whereupon the maximum likelihood estimate and the likelihood ratio function are the main tools. Also, methodology is employed based on the concept of contiguity and related results. In the nonparametric approach, the entities of fundamental importance are unconditional and conditional distribution functions, probability density functions, conditional expectations, and quantiles. Asymp-totic optimal properties are stated for the proposed estimates. In the mixing context, three modes of mixing are entertained but only one, the strong mixing case, is pursued to a considerable extent. Here the approach is exclusively nonparametric. As in the Markovian case, entities estimated are distribution fu nctions, probability density functions and their derivatives, hazard rates, and regression functions. Basic asymptotic optimal properties of the proposed estimates are stated, and precise references are provided. Estimation is proceeded by a discussion of probabilistic results, necessary for statistical inference. It is hoped that this brief and selected review of literature on statistical inference in Markovian and mixing stochastic processes will serve as an introduction to this area of research for those who entertain such an interest. The reason for selecting this particular area of research for a review is that a substantial part of Sidney’s own contributions have been in this area.