On dimension reduction in conditional dependence models

TL;DR

This paper introduces a new approach for estimating the central subspace in conditional dependence models, simplifying inference by decomposing it into marginal and copula components, with an adaptive nonparametric method demonstrating strong performance.

Contribution

It proposes a novel decomposition of the central subspace inference into marginal and copula parts, along with an adaptive nonparametric estimation method that achieves parametric rates.

Findings

The method effectively estimates the central subspace in simulations.

It achieves parametric convergence rates under mild conditions.

Simulation results demonstrate practical applicability.

Abstract

Inference of the conditional dependence structure is challenging when many covariates are present. In numerous applications, only a low-dimensional projection of the covariates influences the conditional distribution. The smallest subspace that captures this effect is called the central subspace in the literature. We show that inference of the central subspace of a vector random variable $\mathbf Y$ conditioned on a vector of covariates $\mathbf X$ can be separated into inference of the marginal central subspaces of the components of $\mathbf Y$ conditioned on $\mathbf X$ and on the copula central subspace, that we define in this paper. Further discussion addresses sufficient dimension reduction subspaces for conditional association measures. An adaptive nonparametric method is introduced for estimating the central dependence subspaces, achieving parametric convergence rates under mild conditions. Simulation studies illustrate the practical performance of the proposed approach.