Higher Moment Estimation for Elliptically-distributed Data: Is it Necessary to Use a Sledgehammer to Crack an Egg?

Abstract: Multivariate elliptically-contoured distributions are widely used for modeling economic and financial data. We study the problem of estimating moment parameters of a semi-parametric elliptical model in a high-dimensional setting. Such estimators are useful for financial data analysis and quadratic discriminant analysis. For low-dimensional elliptical models, efficient moment estimators can be obtained by plugging in an estimate of the precision matrix. Natural generalizations of the plug-in estimator to high-dimensional settings perform unsatisfactorily, due to estimating a large precision matrix. Do we really need a sledgehammer to crack an egg? Fortunately, we discover that moment parameters can be efficiently estimated without estimating the precision matrix in high-dimension. We propose a marginal aggregation estimator (MAE) for moment parameters. The MAE only requires estimating the diagonal of covariance matrix and is convenient to implement. With mild sparsity on the covariance structure, we prove that the asymptotic variance of MAE is the same as the ideal plug-in estimator which knows the true precision matrix, so MAE is asymptotically efficient. We also extend MAE to a block-wise aggregation estimator (BAE) when estimates of diagonal blocks of covariance matrix are available. The performance of our methods is validated by extensive simulations and an application to financial returns.