Kernel order selection by minimum bootstrapped MSE for density weighted averages

Density weighted averages are nonparametric quantities expressed by the expectation of a function of random variables with density weight. It is associated with parametric components of some semiparametric models, and we are concerned with an estimator of these quantities. Asymptotic properties of semiparametric estimators have been studied in econometrics since the end of 1980s and it is now widely recognized that they are n-consistent in many cases. Many of them involve estimates of nonparametric functions such as density and regression function but they are biased estimators for the true functions. Because of this, we typically need to use some bias reduction techniques in the nonparametric estimates for n-consistency of the semiparametric estimators. When we use a kernel estimator, a standard way is to take a higher order kernel function. For density estimation, the higher the kernel order is, the less becomes the bias without changing the order of variance in theory. However, it is also known that higher order kernels can inflate the variance which may cause the result that the mean squared error with very high order kernel becomes larger than that with low order kernel in small sample. This paper proposes to select the bandwidth and kernel order simultaneously by minimizing bootstrap mean squared error for a plug-in estimator of density weighted averages. We show that standard bootstrap does not work at all for bias approximation as in density estimation, but smoothed bootstrap is useful in our problem if suitably transformed.