Nonparametric estimator for the standardized sum using edgeworth expansions

Abstract

Constructing an estimator for functional estimation is one of the problems in statistical inference. In this paper, the problem of constructing an estimator for a studentized sum is considered in the nonparametric set-up. In this set-up, data are used to infer to an unknown quantity while making as few assumptions as possible. This non-smooth functional lack some degree of properties traditionally relied upon in estimation. Smooth functionals converge at the rate of 𝑛−12 while non-smooth functionals converge at the rate slower than 𝑛−12 . This highlights the reason why standard techniques fail to give sharp results. A clear and accurate approximation is obtained by using an approximation that admits cumulant generating function; saddle point approximation. An optimal estimator is obtained using the MiniMax criterion where the lower and upper bounds are constructed. While working in the context of MiniMax estimation, the lower bounds are most important. The MiniMax lower bound is obtained by applying the general lower bound technique based on testing two composite hypotheses. The quality of an estimator is evaluated with the MiniMax risk. Best polynomial approximation of an absolute value function and Hermite polynomials are used to construct an optimal estimator when the means are bounded by a given value $M>0$.