Data based bandwidth selection in kernel density estimation with parametric start via kernel contrasts

Abstract

In contrast to the traditional kernel density estimate which is totally nonparametric, if one has a reasonable parametric guess about the density, it can be used to improve upon the traditional method [Hjort, N. L. and Glad, I. K. (1995). Nonparametric density estimation with a parametric start. Ann. Statist., 23 882–904.]. This semiparametric approach should work in a broad nonparametric neighborhood of a given parametric family. The idea is to multiply the initial parametric guess by a kernel estimate of the correction factor. Since the resulting estimate is clearly not a density, it is corrected by dividing it by its total mass. This correction was missed in the above-mentioned work of Hjort and Glad. This mass corrected version performs better than the uncorrected estimate in the sense of the bias and mean square error. Using the concept of ‘kernel contrast’ [Ahmad, I. A. and Ran, I. S. (1998). Kernel contrasts: a data based method of chosing smoothing parameters in nonparametric density estimation. Unpublished Manuscript.], a totally data based choice of the bandwidth is developed and its finite sample and asymptotic properties are studied. Using this bandwidth, a kernel contrast estimate of the density is given and is shown to perform very well.