Minimum normal approximation error bandwidth selection for averaged derivatives

Density-weighted averaged derivative estimator gives a computationally convenient consistent and asymptotically normally (CAN) distributed estimate of the parametric component of a semiparametric single index model. This model includes some important parametric models as special cases such as linear regression, Logit/Probit, Tobit and Box–Cox and other transformation models. This estimator involves a nonparametric kernel density estimate and thus it faces the problem of bandwidth selection. A reasonable way of bandwidth selection for point estimation is one minimizing the mean squared error. Alternatively, for the purposes of hypothesis testing and confidence interval estimation, we may like to choose it such that it minimizes the normal approximation error. The purpose of this paper is to propose a new bandwidth suitable for these purposes by minimizing the normal approximation error in the tail of exact distribution of the statistics using higher order asymptotic theory of Edgeworth expansion or bootstrap method.