SUFFICIENCY CONDITIONS IN REGULAR MARKOV CHAINS AND CERTAIN RANDOM WALKS
Joe Gani · Biometrika · 1956
For positively regular Markov chains with a finite number of states, transition probabilities of the form are known to admit a sufficient estimator of θ in realizations of the chain starting with a fixed state and consisting of a fixed number of transitions. This paper considers whether transition probabilities of the same form will admit a sufficient estimator of θ in other finite regular, but not positively regular, Markov chains. For chains with an irreducible subset of two or more states, in which a realization starts from a fixed state and consists of a fixed number of transitions, these probabilities are found to admit a maximum-likelihood estimator of the function g(θ)=−λ2'(θ)/Λ1'(θ), which is sufficient and unbiased. There is some difference in chains with an absorbing state, in which realizations start from a fixed state but continue until the absorbing state is reached; in this sequential case, the maximum-likelihood estimator, with the number of transitions in the realization, together provide a sufficient estimator of the function g(θ), which in general is no longer unbiased. We restrict ourselves to the particular case where a certain linear relation is satisfied. This gives rise to some simple stochastic matrices admitting a sufficient estimator of θ, which consist of probabilities of the forms θ and 1−θ; in some of these cases, unbiased sufficient estimators of θ reduce to known results of Girshick, Mosteller & Savage (1946). Some non-regular finite and infinite chains with absorbing states, associated with random walks, whose matrices consist of various patterns of probabilities θ and 1−θ, are also found to admit a sufficient estimator of θ. The paper ends with the examination of such an example, arising in sequential estimation.