Penalized wavelet nonparametric univariate logistic regression for irregular spaced data

Abstract

This paper concerns the study of a non-smooth logistic regression function. The focus is on a high-dimensional binary response case by penalizing the decomposition of the unknown logit regression function on a wavelet basis of functions evaluated on the sampling design. Sample sizes are arbitrary (not necessarily dyadic) and we consider general designs. We study separable wavelet estimators, exploiting sparsity of wavelet decompositions for signals belonging to homogeneous Besov spaces, and using efficient iterative proximal gradient descent algorithms. We also discuss a level by level block wavelet penalization technique, leading to a type of regularization in multiple logistic regression with grouped predictors. Theoretical and numerical properties of the proposed estimators are investigated. A simulation study examines the empirical performance of the proposed procedures, and real data applications demonstrate their effectiveness.

Keywords:

• Nonparametric binary regression
• penalized log-likelihood
• proximal algorithms
• thresholding
• wavelets

Mathematics Subject Classifications:

• 62H12
• MSC 60K35
• MSC 62G08

Acknowledgments

The authors thank the reviewers for their comments on the original version of the paper. Part of this work was completed while A. Antoniadis and I. Gijbels were visiting the Istituto per le Applicazioni del Calcolo ‘M. Picone’, National Research Council, Naples, Italy. I. De Feis acknowledges the INdAM-GNCS 2023 Project ‘Metodi computazionali per la modellizzazione e la previsione di malattie neurodegenerative' and the project 'TAILOR (H2020-ICT-48 GA: 952215)'.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

I. Gijbels gratefully acknowledges financial support from the C16/20/002 project of the Research Fund KU Leuven, Belgium.