Estimation of a Probability Density Function and Its Derivatives

Let X₁, X₂, be independent identically distributed random variables having a common probability density function f. After a so-called kernel class of estimates fₙ of f based on X₁, , Xₙ was introduced by Rosenblatt 7, various convergence properties of these estimates have been studied. The strongest result in this direction was due to Nadaraya 5 who proved that if f is uniformly continuous then for a large class of kernels the estimates fₙ converges uniformly on the real line to f with probability one. For a very general class of kernels, we will show that the above assumptions on f are necessary for this type of convergence. That is, if fₙ converges uniformly to a function g with probability one, then g must be uniformly continuous and the distribution F from which we are sampling must be absolutely continuous with F' (x) = g (x) everywhere. When in addition to the conditions mentioned above, it is assumed that f and its first r + 1 derivatives are bounded, we are able to show how to construct estimates fₙ such that f^ (s) ₙ converges uniformly to f^ (s) at a given rate with probability one for s = 0, 1, , r.