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Abstract:
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Suppose we obtain n i.i.d random vectors O with unknown distribution F. Our goal is to construct minimax estimators for a functional b(F) in a model that places no restriction on F, other than bounds on the roughness of certain density and conditional expectation functions. We consider a general class of functionals that includes: (1) The weighted average treatment effect (2) The marginal mean of a response Y, when Y is missing at random. The functionals in our class have a non-zero semiparametric information bound. However, with the high dimensional data collected in many applications, roughness bounds that are substantively meaningful may yield non-root-n minimax rates of estimation, that are no longer attainable using standard asymptotically linear estimators. We present novel minimax estimators for functionals in our class that are based on higher order U-statistics.
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