Nonparametric Estimation of an Arbitrary Non-smooth Functional Based on Testing a Pair of Composite Hypotheses

Abstract

In this research, an overall MiniMax lower bound (MLB) was derived for the devel- opment of the MiniMax Risk for estimating an arbitrary non-smooth functional, 1 n n P i=1|λi| from an observation Y ∼ N(λ,In) based on testing a pair of composite hypotheses. The Minimax lower bounds and upper bounds are used to quantify the fundamental limits and provide benchmarks for evaluating the performance of any estimator in statistical inference. In nonparametric estimation of statisti- cal functionals, both the lower and upper bounds are constructed. In particular when working in the context of MiniMax estimation, the lower bounds are the most important. The problem of estimating non-smooth functionals shows some properties that are different from those that arise in estimating standard smooth functionals. For these reasons the standard methods fail to give sharp results when used to estimate non-smooth functionals. A pair of priors with a large difference in the expected values of the functional were constructed while making the Chi- square distance between two normal mixtures small. The estimator was developed using the best polynomial approximation, Hermite polynomials and Saddlepoint approximation, and it’s asymptotic properties: bias, variance were derived. The developed estimator was compared with the Nadaraya-Watson and the Modified Nadaraya-Watson estimators. The MSE, biases and confidence interval lengths of the estimators were computed using simulated data. Smaller values of MSE and biases were obtained for the developed estimator. Short confidence interval lengths were also obtained for the developed estimator. The results developed in this research can also be used to solve excess mass.