# zbMATH — the first resource for mathematics

Partial likelihood process and asymptotic normality. (English) Zbl 0632.62088
The notion of partial likelihood, introduced by D. R. Cox and used by R. Gill, K. Dhzaparidze and others for particular cases, is generalized for binary statistical experiments when observations consist of a stochastic process which is a semimartingale with predescribed characteristics.
The idea is to write a formal expression for a likelihood process in spite of the fact that this expression contains as a nuisance parameter a nonobservable process. Nethertheless, it is possible to prove that this partial likelihood process can be used to obtain “nice” estimates of the parameter. In the paper a notion of the partial Hellinger process is presented and a result on asymptotic normality of partial likelihood processes is proved.
Reviewer: Yu.M.Kabanov

##### MSC:
 62M09 Non-Markovian processes: estimation 60G44 Martingales with continuous parameter
Full Text:
##### References:
 [1] Cox, D.R.; Cox, D.R., Partial likelihood, Biometrika, Biometrika, 69-76, (1975) · Zbl 0312.62002 [2] Dellacherie, C.; Meyer, P.A.; Dellacherie, C.; Meyer, P.A., Probabilités et potentiel, I, (), (1976), Hermann Paris [3] Dzhaparidze, K., On asymptotic inference about intensity parameters of a counting process, () · Zbl 0647.62082 [4] Emery, M., Une topologie sur l’espace des semimartingales. Sém. prob. XIII, (), 260-281 [5] Gill, R., Note on product integration, likelihood and partial likelihood for counting processes, () [6] Greenwood, P.; Shiryaev, A.N., Contiguity and the statistical invariance principle, (1985), Gordon and Breach London · Zbl 0659.62029 [7] Hajek, J., Limiting properties of likelihood and inference, () [8] Jacod, J., Calcul stochastique et problèmes de martingales, () · Zbl 0414.60053 [9] Jacod, J., Processus de Hellinger, absolue continuité, contiguité, () [10] Jacod, J., Théorèmes limites pour LES processus, ecole d’été de st flour 1983, () [11] Jacod, J.; Shiryaev, A.N., Limit theorems for stochastic processes, (1987), Springer-Verlag Berlin · Zbl 0635.60021 [12] Lecam, L., Asymptotic methods in statistical decision theory, (1986), Springer Verlag Berlin [13] Liptser, R.S.; Shiryaev, A.N., On the problem of “predictable” criteria of contiguity, (), 384-418 [14] Mémin, J., Espaces de semimartingales et changements de probabilités, Z. für wahrsch., 52, 9-40, (1980) [15] E. Slud, Martingale methods in statistics. To appear. · Zbl 1246.62017 [16] Slud, E., Efficiencies of partial-likelihood-based inferences concerning survival regression models, (1986), Preprint [17] Wong, W., Theory of partial likelihood, Ann. statist., 14, 88-123, (1986) · Zbl 0603.62032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.