Efficiently Estimating the Error Distribution Function in Nonparametric Regression with Responses Missing at Random

• Activity Number: 352
• Type: Contributed
• Date/Time: Tuesday, August 6, 2013 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Justin Chown*+ and Ursula U. Mueller
• Companies: Texas A&M University, Department of Statistics and Texas A&M University, Department of Statistics
• Keywords: nonparametric regression ; local polynomial smoother ; transfer principle ; efficient estimator ; empirical distribution function ; missing at random

Abstract

We consider nonparametric regression models with multivariate covariates and with responses missing at random. We estimate the regression function with a local polynomial smoother. The residual-based empirical distribution function that only uses complete cases, i.e. residuals that can actually be constructed from the data, is shown to be efficient in the sense of H\'ajek and Le Cam. In the proofs we derive, more generally, the efficient influence function for estimating an arbitrary linear functional of the error distribution; this covers the distribution function as a special case. We also show that the complete case residual-based empirical distribution function admits a functional central limit theorem. This is done by applying the transfer principle for complete case statistics developed by Koul et al. (2012), which makes it possible to adapt known results for fully observed data to the missing data case. We conclude with a small simulation study investigating the performance of the complete case residual-based empirical distribution function.