Robust kernel estimator for densities of unknown smoothness

Abstract

Results on non-parametric kernel estimators of density differ according to the assumed degree of density smoothness. A kernel/bandwidth pair that was optimal for a twice differentiable function may not be suitable when the density is piecewise linear. If there is uncertainty about the degree of smoothness, an inappropriate choice may lead to under- or oversmoothing. To examine various possible outcomes we provide asymptotic results on kernel estimation of a continuous density for an arbitrary bandwidth/kernel pair and derive the limit joint distribution of kernel density estimators corresponding to different bandwidths and kernel functions. Using these results, we propose a combined estimator constructed as an optimal linear combination of several estimators with different bandwidth/kernel pairs. Its theoretical properties [Kotlyarova, Y. and Zinde-Walsh, V., 2006, Non- and semi-parametric estimation in models with unknown smoothness. Economics Letters, 93, 379–386] are such that it automatically attains the best possible rate without a priori knowledge of the degree of smoothness. Our Monte-Carlo results confirm the advantages of the combined estimator of density.