Estimation of Non-smooth Functionals in Hilbert Sample Space Using the Edgeworth Expansions

Abstract

An arbitrary non-smooth functional is estimated using a nonparametric set-up. Exploratory data analysis methods are relied on to come up with the functional form for the sample to allow both robustness and optimality to be achieved. An infinite number of parameters are involved and thus the Hilbert sample space is a natural choice. An important step in understanding this problem is the normal means problem, . The basic difficulty of estimating  as defined can be traced back to the non differentiability of the absolute value function, at the origin. Accordingly, constructing an optimal estimator is not easy partly due to the nonexistence of an unbiased estimate of the absolute value function. Therefore, best polynomial approximation was used to smooth the singularity at the origin and then an unbiased estimator for every term in the expansion constructed by use of Hermite polynomials when the averages are bounded by a given constant M > 0 say. The expansion of the Gaussian density function in terms of Hermite polynomials gives a clear and almost accurate estimate that admits cumulant generating function; the Saddle point approximation. Additional precision is obtained by using a higher order Taylor series expansion about the mean resulting in Edgeworth expansion techniques.