ESTIMATING NONQUADRATIC FUNCTIONALS OF A DENSITY USING HAAR WAVELETS

Consider the problem of estimating integral Phi(f), where Phi is a smo oth function and f is a density with given order of regularity s. Spec ial attention is paid to the case Phi(t) = t(3). It has been shown tha t for low values of s the n (-1/2) late of convergence is not achievab le uniformly over the class of objects of regularity s. In fact, a low er bound for this rate is n(-4s/(1+4s)) for 0 < s less than or equal t o 1/4. AS for the upper bound, using a Taylor expansion, it can be see n that it is enough to provide an estimate for the case Phi(x) = x(3). That is the aim of this paper. Our method makes intensive use of spec ial algebraic and wavelet properties of the Haar basis.