N

where -n(x) and ph(x) are nonparametric estimators of n(x) and ph(x) = E(h(X,Y)e I X = x). Note that c is of order n-1/2. Hence XcII(x) is asymptotically equivalent to the nonparametric estimator X Nevertheless, it leads to a better estimator for E[h(X,Y)]. Under appropriate assumptions, c has the expansion

with d = E[Zph(X)X/n(X)a2(X)] = E[h(X,Y)a-2(X)Xe]. Using the expansion for the weighted least squares estimator ß, we see that

Using this and the stochastic expansion of the nonparametric estimator x, we obtain that the estimators H1 - c and H2 - c have influence functions ^ = ^n and are therefore efficient by Section 2. Of course, H2 - c is the fully imputed estimator based on xii. Both H1 - c and H2 - c are better than the partially imputed estimators H1 based on the estimator x, and H1 - (1 - Z}c based on the estimator xii.

Simpler estimators are possible for certain functions h, such as h(x, y} = y, which is the function usually treated in the literature. Since E(Y | X} = $X, we can use the fully imputed estimator i)X, with X = n £n= \ Xi. As smooth function of the two efficient estimators $ and X, the estimator dX is efficient for E(Y I X}. Matloff [Mat81] has recommended an estimator of this form, but with a simpler, in general inefficient, estimator for

Achnowledgment

Anton Schick was supported in part by NSF Grant DMS 0072174.

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