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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

62F12 Asymptotic properties of parametric estimators
62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
Full Text: DOI
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