ON ESTIMATION OF GENERALIZED DENSITIES

This paper is concerned with the problem of estimating a generalized density of type f(t) = (1 - alpha)g(t) + alpha-SIGMA(i is-an-element-of I)p(i)delta(t - a(i)) (delta denotes the Dirac delta) corresponding to a probability distribution function without a singular part. We propose two natural classes of estimates and obtain consistency results with respect to suitable norms. Our approach could be useful to provide an alternative model for those problems (simulation, smoothed bootstrap) in which density estimates are used as auxiliary tools to draw artificial samples. In particular, we suggest the possibility of considering a sort of mixed bootstrap, placed in an intermediate position between the two usual versions (smoothed and unsmoothed) of this technique. Certain interesting sequences of integer-valued random variables [called rare numbers by E. Key (1991)] arise, in a natural way, in the consistency proofs.