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

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
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