Asymptotic expansion of the log-likelihood function based on stopping times defined on a Markov process

Abstract

Consider the parameter space Θ which is an open subset of ℝ^k,k≧1, and for each θ∈Θ, let the r.v.′sY _n ,n=0, 1, ... be defined on the probability space (X,A,P_θ) and take values in a Borel setS of a Euclidean space. It is assumed that the process {Y _n },n≧0, is Markovian satisfying certain suitable regularity conditions. For eachn≧1, let υ _n be a stopping time defined on this process and have some desirable properties. For 0 < τ _n → ∞ asn→∞, set$\theta _{\tau _n } = \theta + h_n \tau _n^{ - 1/2}$h _n →h ∈R^k, and consider the log-likelihood function$\Lambda _{\nu _n } (\theta )$ of the probability measure$\tilde P_{n,\theta _{r_n } }$ with respect to the probability measure$\tilde P_{n,\theta }$. Here$\tilde P_{n,\theta }$ is the restriction ofP_θ to the σ-field induced by the r.v.′sY₀,Y₁, ...,$Y_{\nu _n }$. The main purpose of this paper is to obtain an asymptotic expansion of$\Lambda _{\nu _n } (\theta )$ in the probability sense. The asymptotic distribution of$\Lambda _{\nu _n } (\theta )$, as well as that of another r.v. closely related to it, is obtained under both$\tilde P_{n,\theta }$ and$\tilde P_{n,\theta _{r_n } }$.