Homogeneity in Regression

TL;DR

This paper introduces CARDS, a new clustering algorithm for high-dimensional regression that exploits coefficient homogeneity, improving estimation accuracy by leveraging low-dimensional structures like homogeneity and sparsity.

Contribution

The paper proposes a novel data-driven segmentation method called CARDS to explore homogeneity in regression coefficients, with theoretical analysis and practical applications.

Findings

CARDS achieves better estimation accuracy for homogeneous parameters.

The asymptotic normality of CARDS estimator is established.

Combining homogeneity and sparsity exploration enhances efficiency.

Abstract

This paper explores the homogeneity of coefficients in high-dimensional regression, which extends the sparsity concept and is more general and suitable for many applications. Homogeneity arises when one expects regression coefficients corresponding to neighboring geographical regions or a similar cluster of covariates to be approximately the same. Sparsity corresponds to a special case of homogeneity with a known atom zero. In this article, we propose a new method called clustering algorithm in regression via data-driven segmentation (CARDS) to explore homogeneity. New mathematics are provided on the gain that can be achieved by exploring homogeneity. Statistical properties of two versions of CARDS are analyzed. In particular, the asymptotic normality of our proposed CARDS estimator is established, which reveals better estimation accuracy for homogeneous parameters than that without homogeneity exploration. When our methods are combined with sparsity exploration, further efficiency can be achieved beyond the exploration of sparsity alone. This provides additional insights into the power of exploring low-dimensional strucuture in high-dimensional regression: homogeneity and sparsity. The newly developed method is further illustrated by simulation studies and applications to real data.