### 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.′s Y_{ n}, n=0, 1, ... be defined on the probability space (X, A, P_{ θ}) and take values in a Borel set S of a Euclidean space. It is assumed that the process {Y_{ n} }, n≧0, is Markovian satisfying certain suitable regularity conditions. For each n≧1, let υ_{ n} be a stopping time defined on this process and have some desirable properties. For 0 < τ_{ n} → ∞ as n→∞, set {Mathematical expression} h_{ n} →h ∈R^{ k}, and consider the log-likelihood function {Mathematical expression} of the probability measure {Mathematical expression} with respect to the probability measure {Mathematical expression}. Here {Mathematical expression} is the restriction of P_{ θ} to the σ-field induced by the r.v.′s Y_{ 0}, Y_{ 1}, ..., {Mathematical expression}. The main purpose of this paper is to obtain an asymptotic expansion of {Mathematical expression} in the probability sense. The asymptotic distribution of {Mathematical expression}, as well as that of another r.v. closely related to it, is obtained under both {Mathematical expression} and {Mathematical expression}.

Original language | English (US) |
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Pages (from-to) | 21-38 |

Number of pages | 18 |

Journal | Annals of the Institute of Statistical Mathematics |

Volume | 31 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 1979 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Mathematics(all)