A Covariance Matrix Estimator Based on the Singular Wishart Likelihood

• Activity Number: 249
• Type: Contributed
• Date/Time: Monday, July 30, 2012 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Samprit Banerjee*+ and Stefano Monni
• Companies: Weill Cornell Medical College and Weill Cornell Medical College
• Keywords: covariance matrix ; singular sample covariance matrix ; marginal likelihood ; singular wishart

Abstract

In recent years there has been a surge of applications with high dimensional data in genetics, finance, internet portals etc. Often times one is interested in estimating the high dimensional covariance matrix when the data collected is insufficient, in other words, p >> n where p denotes the dimension of the covariance matrix and n denotes the number of samples. The usual estimator, the sample covariance matrix 1/n X'X is known to be a poor estimator even when p is moderately large and singular when p>>n. We consider the class of orthogonally invariant estimators and derive an approximation to the marginal likelihood (with respect to eigenvalues) for this problem. The approximation here is based on large p and not on the conventional limit of large n. We also derive the Maximum Likelihood Estimator based on this approximate marginal likelihood. We will show via simulations that the maximum likelihood estimator of the approximate marginal likelihood outperforms the sample covariance matrix using different loss functions.