Missing, Modified, and Large-p-small-n Data in Nonparametric Curve Estimation

Nonparametric curve estimation, which makes no assumptions about shape of estimated functions, is one of the main pillars of the modern statistical science. It is used when no adequate parametric or semi-parametric model is available. Asymptotic results on adaptive estimation of nonparametric curves, under both traditional and shrinking minimaxes, are presented. The latter approach allows us to explain the phenomenon of superefficiency when a function can be estimated with a rate faster than the minimax one. Resent results on sequential nonparametric estimation, which yields an assigned risk with minimal average stopping time, are also presented. Then it is explained how the theory can be used in practically important cases of missing data, modified data in survival analysis, and Big Data. Wavelet applications in the analysis of microarrays and fMRI illustrate feasibility of the proposed nonparametric estimation.