Abstract

The best mean square error that the classical kernel density estimator achieves if the kernel is non-negative and f has only two continuous derivatives, is of the order of [special characters omitted]. If negative kernels are allowed, then this rate can be improved depending on the smoothness of f and the order of the kernel.

Abramson and others modified the classical kernel estimator, assumed non-negative, by allowing the bandwidth h_n to depend on the data. The last and best result in the literature is Hall, Hu and Marron who show that under suitable assumptions on a non-negative kernel K and the density f, |fˆ_n( t) − f(t)| = O _P([special characters omitted]) for fixed t. The main result of this thesis states that [special characters omitted] |fˆ_n(t) − f(t)| = O_P(([special characters omitted])^4/9) where D_n and fˆ_n(t) are purely data driven and D_n can be taken as close as desired to the set { t : f(t) > 0}. This rate is best possible for estimating a density in the sup norm. The data driven fˆ_n(t) and D_n have 'ideal' counterparts that depend on f, and for the ideal estimator, slightly sharper results are proven.