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

Authors:	M G Akritas G G Roussas

Institution:	(1) University of Wisconsin, Madison;(2) University of Patras, Patras, Greece

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.′sY ₀,Y ₁, ..., $Y_{\nu _n }$ . 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 $\tilde P_{n,\theta _{r_n } }$ . This research was supported by the National Science Foundation, Grant MCS77-09574. Research supported by the National Science Foundation, Grant MCS76-11620.

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