A Note on Choosing the Threshold for Large Covariance Estimations in Factor Models

TL;DR

This paper proposes a simple plug-in thresholding method for estimating large covariance matrices in factor models, improving the minimax rate of convergence and relaxing sample size conditions compared to previous work.

Contribution

It introduces a new threshold selection method based on a plug-in approach, extending the minimax rate of convergence for large covariance matrix estimation in factor models.

Findings

The proposed method achieves the minimax rate of convergence.

It relaxes the sample size condition from n=o(p log p) to n=o(p^2 log p).

The threshold is motivated by the tuning parameter in the lasso literature.

Abstract

This note shows that for i.i.d. data, estimating large covariance matrices in factor models can be casted using a simple plug-in method to choose the threshold: $$ \mu_{jl}=\frac{c_0}{\sqrt{n}}\Phi^{-1}(1-\frac{\alpha}{2p^2})\sqrt{\frac{1}{n}\sum_{i=1}^n\hat u_{ji}^2\hat u_{li}^2}.$$ This is motivated by the tuning parameter suggested by Belloni et al. (2012) in the lasso literature. It also leads to the minimax rate of convergence of the large covariance matrix estimator. Previously, the minimaxity is achievable only when $n=o(p\log p)$ by Fan et al. (2013), and now this condition is weakened to $n=o(p^2\log p)$. Here $n$ denotes the sample size and $p$ denotes the dimension.