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

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.