Nonsingular subsampling for regression S estimators with categorical predictors

Abstract

Simple random subsampling is an integral part of S estimation algorithms for linear regression. Subsamples are required to be nonsingular. Usually, discarding a singular subsample and drawing a new one leads to a sufficient number of nonsingular subsamples with a reasonable computational effort. However, this procedure can require so many subsamples that it becomes infeasible, especially if levels of categorical variables have low frequency. A subsampling algorithm called nonsingular subsampling is presented, which generates only nonsingular subsamples. When no singular subsamples occur, nonsingular subsampling is as fast as the simple algorithm, and if singular subsamples do occur, it maintains the same computational order. The algorithm works consistently, unless the full design matrix is singular. The method is based on a modified LU decomposition algorithm that combines sample generation with solving the least squares problem. The algorithm may also be useful for ordinary bootstrapping. Since the method allows for S estimation in designs with factors and interactions between factors and continuous regressors, we study properties of the resulting estimators, both in the sense of their dependence on the randomness of the sampling and of their statistical performance.