Numerical Implementation of the QuEST Function

TL;DR

This paper presents a practical numerical method for implementing the QuEST function, enabling accurate estimation of covariance matrix eigenvalues in high-dimensional settings, supported by an efficient algorithm and simulation results.

Contribution

It introduces a detailed six-step numerical implementation and an analytical Jacobian computation for the QuEST function, facilitating its use in large-dimensional covariance estimation.

Findings

Effective numerical implementation of QuEST function demonstrated

Algorithm accurately computes Jacobian for optimization

Simulations confirm the method's robustness and accuracy

Abstract

This paper deals with certain estimation problems involving the covariance matrix in large dimensions. Due to the breakdown of finite-dimensional asymptotic theory when the dimension is not negligible with respect to the sample size, it is necessary to resort to an alternative framework known as large-dimensional asymptotics. Recently, Ledoit and Wolf (2015) have proposed an estimator of the eigenvalues of the population covariance matrix that is consistent according to a mean-square criterion under large-dimensional asymptotics. It requires numerical inversion of a multivariate nonrandom function which they call the QuEST function. The present paper explains how to numerically implement the QuEST function in practice through a series of six successive steps. It also provides an algorithm to compute the Jacobian analytically, which is necessary for numerical inversion by a nonlinear optimizer. Monte Carlo simulations document the effectiveness of the code.