Deciding the dimension of effective dimension reduction space for functional and high-dimensional data

TL;DR

This paper introduces new sequential chi-squared testing procedures to determine the dimension of the effective dimension reduction space in functional and high-dimensional data regression models, supported by theory, simulations, and real data.

Contribution

It proposes novel sequential chi-squared tests for EDR space dimension determination applicable to functional and high-dimensional data, extending existing methods.

Findings

Proposed procedures accurately determine EDR dimension in simulations.

The methods are theoretically justified and validated with real data examples.

Applicable to both functional and high-dimensional multivariate data.

Abstract

In this paper, we consider regression models with a Hilbert-space-valued predictor and a scalar response, where the response depends on the predictor only through a finite number of projections. The linear subspace spanned by these projections is called the effective dimension reduction (EDR) space. To determine the dimensionality of the EDR space, we focus on the leading principal component scores of the predictor, and propose two sequential $\chi^2$ testing procedures under the assumption that the predictor has an elliptically contoured distribution. We further extend these procedures and introduce a test that simultaneously takes into account a large number of principal component scores. The proposed procedures are supported by theory, validated by simulation studies, and illustrated by a real-data example. Our methods and theory are applicable to functional data and high-dimensional multivariate data.