Average Derivative Estimation Under Measurement Error

In this paper, we derive the asymptotic properties of average derivative estimators when the regressors are contaminated with classical measurement error and the density of this error is unknown. Average derivatives of conditional mean functions are used extensively in economics and statistics, most notably in semiparametric index models. As well as ordinary smooth measurement error, we provide results for supersmooth error distributions. This is a particularly important class of error distribution as it includes the popular Gaussian density. We show that under this ill-posed inverse problem, despite using nonparametric deconvolution techniques and an estimated error characteristic function, we are able to achieve a \sqrt{n} rate of convergence for the average derivative estimator. Interestingly, if the measurement error density is symmetric, the asymptotic variance of the average derivative estimator is the same irrespective of whether the error density is estimated or not.