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Theory of partial likelihood. (English) Zbl 0603.62032
This paper develops the asymptotic theory: consistency and asymptotic normality for estimators of a parameter \(\theta\) based upon partial likelihood or conditional likelihood in a rather general setting. It includes most of the results obtained in this area, among them we can quote J. D. Kalbfleisch and D. A. Sprott [J. R. Stat. Soc., Ser. B 32, 175-208 (1970; Zbl 0205.459)], D. R. Cox [Biometrika 62, 269-276 (1975; Zbl 0312.62002)], B. G. Lindsay [Philos. Trans. R. Soc. Lond., Ser. A 296, 639-665 (1980; Zbl 0434.62028); Biometrika 69, 503-512 (1982; Zbl 0498.62007); Ann. Stat. 11, 486-497 (1983; Zbl 0583.62024)].
The theory is of special interest when the partial likelihood involves only the parameter \(\theta\) while the other part or the full likelihood involve a nuisance parameter \(\eta\) (of finite or infinite dimension). The approach to the consistency problem follows Doob and Wald’s one which avoids differentiability and uniqueness conditions. The treatment of asymptotic normality for the partial likelihood MLE is an extension of P. Billingsley, Statistical inference for Markov processes. (1961; Zbl 0106.342), and following papers, see P. Hall and C. C. Heyde, Martingale limit theory and its application. (1980; Zbl 0462.60045), or I. V. Basawa and B. L. S. Prakasa Rao, Statistical inference for stochastic processes. (1980; Zbl 0448.62070), using the martingale differences structure of conditional scores.
A special attention is devoted to the problem of loss of efficiency. The investigation is supported by Bahadur and Hajek’s theories; it is shown that the minimal Fisher information (Lindsay, 1980) provides a lower bound for the asymptotic variance of regular estimates. The last section gives various examples such as missing values in AR processes or more general processes, or the proportional hazard model (without censoring).
Reviewer: J.Deshayes

MSC:
62F12 Asymptotic properties of parametric estimators
62A01 Foundations and philosophical topics in statistics
62M09 Non-Markovian processes: estimation
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