Smoothness Adaptive AverageDerivative Estimation

Many important models, such as index models widely used in limiteddependent variables, partial linear models and nonparametric demand studiesutilize estimation of average derivatives (sometimes weighted) of theconditional mean function. Asymptotic results in the literature focus onsituations where the ADE converges at parametric rates (as a result ofaveraging); this requires making stringent assumptions on smoothness of theunderlying density; in practice such assumptions may be violated. We extendthe existing theory by relaxing smoothness assumptions. We consider boththe possibility of lack of smoothness and lack of precise knowledge of degreeof smoothness and propose an estimation strategy that produces the bestpossible rate without a priori knowledge of degree of density smoothness. Thenew combined estimator is a linear combination of estimators correspondingto different bandwidth/kernel choices that minimizes the trace of the part ofthe estimated asymptotic mean squared error that depends on the bandwidth.Estimation of the components of the AMSE, of the optimal bandwidths,selection of the set of bandwidths and kernels are discussed. Monte Carloresults for density weighted ADE confirm good performance of the combinedestimator.