Estimating a real parameter in a class of semiparametric models. (English) Zbl 0665.62034

van der Vaart, A. W.

Ann. Stat. 16, No. 4, 1450-1474 (1988).

The author studies an estimation problem in semiparametric models where for a fixed value of a finite-dimensional parameter there exists a sufficient statistic for the nuisance parameter. Namely, if \(X_ 1,X_ 2,...,X_ n\) are independent random elements with respective densities \(p_ j(\cdot,\theta,\eta)\), where \(\theta\) is the parameter of interest while \(\eta\) is the nuisance parameter, then it is assumed that \[ p_ j(\cdot,\theta,\eta)=h_ j(\cdot,\theta)g(\psi_ j(\cdot,\theta),\theta,\eta) \] and \(\psi (X_ j,\theta)\) has density g(\(\cdot,\theta,\eta)\) w.r.t. the measures \(v_{\theta}\) for some functions \(h_ j\) and \(\psi_ j\). One of the basic assumptions concerning the score functions is that projecting on the set of nuisance scores is equivalent to taking conditional expectations given the sufficient statistics \(\psi\).
An estimator sequence \(\{T_ n\}\) is said to be asymptotically efficient for \(\theta\) if \[ \sqrt{n}(T_ n-\theta)=n^{- 1/2}\sum^{n}_{j=1}\tilde I_ n^{-1}(\theta,\eta)\tilde l_{nj}(X_ j,\theta,\eta)+o_{P_{\theta \eta}}(1), \] where \(\tilde I_ n\) are normalizing factors and \(\tilde l_{nj}\) is a naturally defined efficient score function. The efficient estimator \(T_ n\) is obtained by the one-step method: \[ T_ n={\hat \theta}_ n+(1/n)\sum^{n}_{j=1}I_ n^{-1}({\hat \theta}_ n)\hat l_{nj}(X_ j,{\hat \theta}_ n), \] where \({\hat \theta}{}_ n\) is a \(\sqrt{n}\)-consistent initial estimator and \(\hat l_{nj}\) is an estimator of the efficient score function. The estimator is in fact optimal in the sense of the convolution and local asymptotic minimax theorem. The paper is supplemented by a number of examples of mixture models, where a useful concept of local completeness is used.

Reviewer: T.Bednarski

MSC:

62F12	Asymptotic properties of parametric estimators
62F35	Robustness and adaptive procedures (parametric inference)
62F10	Point estimation
62G05	Nonparametric estimation
62G20	Asymptotic properties of nonparametric inference

Keywords:

asymptotically normal sequence of estimators; conditional maximum likelihood estimation; asymptotic efficient estimation; adaptation; semiparametric models; sufficient statistic; nuisance parameter; score functions; conditional expectations; convolution; local asymptotic minimax theorem; mixture models; local completeness