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Statistica Sinica 32 (2022), 673-694

HIGH-DIMENSIONAL VARYING INDEX COEFFICIENT
QUANTILE REGRESSION MODEL

Jing Lv and Jialiang Li

Southwest University and National University of Singapore

Abstract: Statistical learning is evolving quickly, with increasingly sophisticated models seeking to incorporate the complicated data structures from modern scientific and business problems. Varying-index coefficient models extend varying-coefficient models and single-index models for semiparametric regressions. This new class of model offers greater flexibility in terms of characterizing complicated nonlinear interaction effects in a regression analysis. To safeguard against outliers and extreme observations, we consider a robust quantile regression approach to estimate the model parameters. High-dimensional loading parameters are allowed in our development, under reasonable theoretical conditions. Thus, we propose a regularized estimation procedure to select the significant nonzero loading parameters, identify linear functions in varying-index coefficient models, and consistently estimate the parametric and nonparametric components. Under some technical assumptions, we show that the proposed procedure is consistent in terms of variable selection and linear function identification, and that the proposed parameter estimation enjoys the oracle property. Extensive simulation studies are carried out to assess the finite-sample performance of the proposed method. We illustrate our methods using an example based on New Zealand workforce data.

Key words and phrases: High-dimensional data, penalty, quantile regression, semi-parametric regression, varying index coefficient model.

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