Confidence intervals for nonparametric regression functions with missing data: Multiple design case

Abstract

This paper considers two estimators of θ = g(x) in a nonparametric regression model Y = g(x) + ε(x ∈ (0, 1) p ) with missing responses: Imputation and inverse probability weighted estimators. Asymptotic normality of the two estimators is established, which is used to construct normal approximation based confidence intervals on θ.

Empilrical likelihood for non-parametric regression models with missing responses: Multiple design case

“…Let U ∼ N (0, 1) and u α satisfy P (|U | ≤ u α ) = 1 − α. Then normal approximation based confidence intervals on θ = g(x) with asymptotically coverage probability 1 − α based on nonparametric regression and inverse probability weighted imputation methods are constructed respectively as (see [7])…”

A simple approach to construct confidence bands for a regression function with incomplete data

“…. Some work has been done for the simpler problem of con- Qin et al (2014) as well as Lei and Qin (2011). However, to the best of our knowledge, a long-standing problem in constructing uniform confidence bands for the regression function m(x), over compact sets, involves the situation where the response variable Y may be missing at random and the function p(x) in (2), the density function f (x) of X, and the conditional variance σ 2 0 (x) = E[Y 2 |X = x] − m 2 (x) are all completely unknown.…”