Skewing methods for two-parameter locally parametric density estimation

A 'skewing' method is shown to effectively reduce the order of bias of loca lly parametric estimators, and at the same time retain positivity propertie s. The technique involves first calculating the usual locally parametric ap proximation in the neighbourhood of a point x' that is a short distance fro m the place x where we wish to estimate the density, and then evaluating th is approximation at x. By way of comparison, the usual locally parametric a pproach takes x' = x. In our construction, x' - x depends in a very simple way on the bandwidth and the kernel, and not at all on the unknown density. Using skewing in this simple form reduces the order of bias from the squar e to the cube of bandwidth; and taking the average of two estimators comput ed in this way further reduces bias, to the fourth power of bandwidth. On t he other hand, variance increases only by at most a moderate constant facto r.