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

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 semi parametric 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 Unpublished Manuscript.], a totally data based choice of the bandwidth is developed and its finite sample and asymptotic properties are studied.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.