Theoretical properties of Cook's PFC dimension reduction algorithm for linear regression

TL;DR

This paper provides a theoretical analysis of Cook's Principal Fitted Components (PFC) algorithm for linear regression, establishing conditions for estimator consistency and comparing its performance to principal components.

Contribution

The paper introduces new theoretical insights into the properties of PFC, including consistency conditions and performance comparisons with PCA, using advanced mathematical techniques.

Findings

PFC estimators can be $\

Under certain conditions, PFC achieves $\

Compared to PCA, PFC should perform better under Cook's model.

Abstract

We analyse the properties of the Principal Fitted Components (PFC) algorithm proposed by Cook. We derive theoretical properties of the resulting estimators, including sufficient conditions under which they are $\sqrt{n}$-consistent, and explain some of the simulation results given in Cook's paper. We use techniques from random matrix theory and perturbation theory. We argue that, under Cook's model at least, the PFC algorithm should outperform the Principal Components algorithm.