Smoothness adaptive average derivative estimation

Abstract

Many important models utilize estimation of average derivatives of the conditional mean function. Asymptotic results in the literature on density weighted average derivative estimators (ADE) focus on convergence at parametric rates; this requires making stringent assumptions on smoothness of the underlying density; here we derive asymptotic properties under relaxed smoothness assumptions. We adapt to the unknown smoothness in the model by consistently estimating the optimal bandwidth rate and using linear combinations of ADE estimators for different kernels and bandwidths. Linear combinations of estimators (i) can have smaller asymptotic mean squared error (AMSE) than an estimator with an optimal bandwidth and (ii) when based on estimated optimal rate bandwidth can adapt to unknown smoothness and achieve rate optimality. Our combined estimator minimizes the trace of estimated MSE of linear combinations. Monte Carlo results for ADE confirm good performance of the combined estimator. Copyright (C) The Author(s). Journal compilation (C) Royal Economic Society 2010.

Normal state charge dynamics ofFe1.06Te0.88S0.14superconductor probed with infrared spectroscopy

“…Apart from the vertical offset 29 , within the noise level the ratio is a straight line, which indicates the absence of field-induced effects in Fe 1.06 Te 0.88 S 0.14 . This is in contrast with BaFe 2−x Co x As 2 with T c = 22 K (optimally doped member of the 122 family), where clear field-induced changes have been observed caused by the suppression of the superconducting gap 30 . We speculate that field-induced changes in Fe 1.06 Te 0.88 S 0.14 are not observed because either they are below the detection limit of our experiment, or because the superconductiong gap is outside of our frequency window.…”

Averaging of an Increasing Number of Moment Condition Estimators

“…Alternatively, one could choose to retain only those estimators with t-statistic, say, that exceeds some predetermined level like one. Alternatively, if G nj is continuously differentiable with respect to θ, one could also use results in Rilstone, Srivastava, and Ullah (1996); Rilstone and Ullah (2005) to estimate J n and the weights by means of minimizing an estimate of the proposed estimator's mean square error (see e.g., Schafgans and Zinde-Walsh, 2010). The theoretical justification of the latter is left for future research.…”

“…In the nonparametric literature, Gray and Schucany (1972) and Bierens (1987) have proposed jackknife estimators that combine different kernel smoothers in order to reduce bias. Similarly, Zinde-Walsh (2006, 2007) and Schafgans and Zinde-Walsh (2010) have proposed combining kernel smoothers calculated with different bandwidths and kernel functions to construct robust estimators of densities and density-weighted average derivatives respectively. In additive nonparametric regression, integration, or averaging, has been shown to improve rates of convergence and to eliminate nuisance parameters (see e.g., Linton and Nielsen, 1995).…”

Averaging of moment condition estimators

“…In the nonparametric literature, Gray and Schucany (1972) and Bierens (1987) have proposed jacknife estimators that combine different kernel smoothers in order to reduce bias. Similarly, Kotlyarova andZinde-Walsh (2006, 2007) and Schafgans and Zinde-Walsh (2010) have proposed combining kernel smoothers calculated with different bandwidths and kernel functions to construct robust estimators of densities and density-weighted average derivatives respectively.…”

“…Alternatively, if G nj is continuously differentiable with respect to θ, one could also use results in Rilstone, Srivastava, and Ullah (1996); Rilstone and Ullah (2005) to estimate J n and the weights by means of minimizing an estimate of the proposed estimator's mean square error, see e.g. Schafgans and Zinde-Walsh (2010). The theoretical justification of the latter is left for future research.…”