A Fourier inversion method for the estimation of a density and its derivatives

Abstract

During the last two decades a variety of methods have been developed for the problem of estimation of unknown density f wrt Lebesgue measure and its vth dervative g(= f(v)) using i.i.d random variables X1, …, Xn when X1 ∼ f. For example, see Wegman (1972). In almost all the papers on the estimation of f(x) or g(x), various authors assumed the existence of derivatives of f of order r(>v) at x to obtain rates for the mean-square convergences and other desirable properties for their estimators. Here it is shown that ifthen estimators gˇ(x) can be constructed for which E[(gˇ(x)-g(x))2] = O(n−(2a-δ)/(2a+2v+1)) for anygiveng δ > 0. Simiar statements hold for almost sure convergence of gˇ(x). It can also be shown that (gˇ(x1), gˇ(x2)) is asymptotically bivariate normal under certain conditions for x1, ≠ x2. If (Al) is satisfied with a ≦ l, then our estimators have all the desirable properties while other methods are not applicable in this situation since they require differentiability conditions on g. (For example, see Susarla and Kumar (1975) and its references.) Our estimators are defined by using the inversion theorem for some absolutely integrable characteristic functions. The motivation for our estimators is given in O'Bryan and Susarla (1975, 76) and Susarla and O'Bryan (1975).