Nonsingular subsampling for regression S estimators with categorical predictors

Koller, Manuel; Stahel, Werner A. (2017). Nonsingular subsampling for regression S estimators with categorical predictors. Computational Statistics, 32(2), pp. 631-646. Springer 10.1007/s00180-016-0679-x

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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 non-
singular 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 nonsingu-
lar 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.

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