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Asymptotic efficiency in estimation with conditional moment restrictions. (English) Zbl 0618.62040
Suppose data are i.i.d. according to some unknown $$F\in {\mathcal F}$$ and suppose $$\theta$$ is a parameter, real or vector valued, defined on $${\mathcal F}$$. Let (z,$$\theta)$$ be an estimating function, a function of parameter as well as of observations. As in regression, the restriction is that the conditional expectation of (z,$$\theta)$$ is zero at a particular point $$\theta_ 0$$ which is to be estimated.
The author first proves the result that if F is a multinomial distribution with known finite support then the lower bound $$\Lambda_ 0$$, for the variance-covariance matrix of the estimator is obtained from the information matrix. This lower bound is w.r.t. the partial order defined by non-negative definiteness of the difference between the variance-covariance matrices of the competing estimators.
It is shown that this bound depends only on certain conditional moments and not on the support of the multinomial F. Since a general F can be approximated by a multinomial distribution the lower bound $$\Lambda_ 0$$ provides the lower bound in general.
Reviewer: B.K.Kale

##### MSC:
 62F12 Asymptotic properties of parametric estimators 62G20 Asymptotic properties of nonparametric inference 62G05 Nonparametric estimation
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