Statistical Methods in Markov Chains

TLDR

The paper reviews statistical inference for finite Markov chains, reflecting the author's interests and citing key works such as Grenander, Bartlett, Fortet, and the author's monograph. The study surveys mathematical methods for inferring transition probabilities from a single long observation of a finite Markov chain and indicates how these methods extend to arbitrary state spaces or continuous time. It covers Whittle's formula, chi‑square and maximum‑likelihood estimation, parameter estimation, and multiple chains, and provides proofs of Whittle's formula and limit theory for chi‑square methods. The paper presents results with literature references rather than full proofs and briefly shows applicability to general state spaces or continuous time. The author thanks Paul Meier for helpful discussions.

Abstract

This paper is an expository survey of the mathematical aspects of statistical inference as it applies to finite Markov chains, the problem being to draw inferences about the transition probabilities from one long, unbroken observation $\{x_1, x_2, \cdots, x_n\}$ on the chain. The topics covered include Whittle's formula, chi-square and maximum-likelihood methods, estimation of parameters, and multiple Markov chains. At the end of the paper it is briefly indicated how these methods can be applied to a process with an arbitrary state space or a continuous time parameter. Section 2 contains a simple proof of Whittle's formula; Section 3 provides an elementary and self-contained development of the limit theory required for the application of chi-square methods to finite chains. In the remainder of the paper, the results are accompanied by references to the literature, rather than by complete proofs. As is usual in a review paper, the emphasis reflects the author's interests. Other general accounts of statistical inference on Markov processes will be found in Grenander [53], Bartlett [9] and [10], Fortet [35], and in my monograph [18]. I would like to thank Paul Meier for a number of very helpful discussions on the topics treated in this paper, particularly those of Section 3.