Some Asymptotic Results and Exponential Approximation in Semi-Markov Models

Abstract

Consider a continuous-time parameter semi-Markov process {X t , t ≥ 0} with the random variables involved defined on the probability space (Ω,A, P θ ) and taking values in the finite set {1,2,…, m}; θ is a k-dimensional parameter belonging in an open set θ the parameter space. For T (greaterthan) 0 let P T, θ be the restriction of P θ to the б -field generated by the random variables {X t , 0 ≤ t ≤ T}. Under suitable regularity conditions, the log-likelihood ratio, log(dPT,θ* /dPT,θ), exists for all θ and θ*, and for parameter points θ* tending to θ at a specified rate, as T (ARROW) ∞, the corresponding log-likelihood ratio assumes a certain expansion in the probability sense. The asymptotic distribution of the log-likelihood ratio is also determined, as is the asymptotic distribution of a k-dimensional random vector closely related to the log-likelihood ratio in the sense of probability. These results hold both under a sequence of probability measures depending on a fixed parameter point, as well as under a sequence of probability measures contiguous to the former sequence. Also, it is shown that, in the neighborhood of a parameter point 0, the underlying family of probability measures may be approximated, in the sup-norm sense, by an exponential family of probability measures. These results may be used for carrying out statistical inference with asymptotically optimal properties.