Walsh-average based variable selection for varying coefficient models

A shrinkage-type variable selection procedure for varying coefficient models is routinely established in the least-squares (LS) framework. Although the LS method has favorable properties for a large class of error distributions, it will break down if the error variance is infinite and is adversely affected by outliers and heavy-tail distributions. To overcome these issues, we propose a robust shrinkage method termed regularized Walsh-average (RWA) that can construct robust nonparametric variable selection and robust coefficient estimation simultaneously. Theoretical analysis reveals RWA works beautifully, including consistency in variable selection and oracle property in estimation, even when error variance is infinite. More important property is that when error variance is finite, compared with the LS based estimators, the asymptotic relative efficiency of the new estimator is at least 0.8896, a relatively high level. Furthermore, a robust BIC-type criterion, which can identify the true model consistently, is suggested for shrinkage parameter selection. Numerical studies also confirm our theories.