Sequential Analysis of Nonparametric Heteroscedastic Regression with Missing Responses

Abstract

Regression for data with randomly missed responses is a well-known and complicated statistical problem. This article, for the first time in the literature, explores the asymptotic theory of sharp minimax sequential estimation of a regression function for two classical settings. The former is when the design of predictors is random and an expected stopping time (or its moment) is bounded, and the latter is when the sample size is fixed and predictors can be chosen sequentially to attenuate effects of heteroscedasticity and missing responses. For the former setting it is shown that sequential estimation cannot outperform a design with a fixed sample size. This conclusion expands the famous single-parameter result of Anscombe (1952) upon nonparametric regression (infinite-dimensional parameter) with missing data. For the latter setting a sequential design of predictors is proposed that allows the statistician to match performance of a sharp minimax oracle-estimator that knows all nuisance functions and parameters, including the scale function, the conditional probability of missing the response given the predictor, and smoothness of estimated regression. A numerical study is presented.