Higher Moment Estimation for Elliptically Distributed Data: Is It Necessary to Use a Sledgehammer to Crack an Egg?

Motivated by applications in financial data analysis and quadratic discriminant analysis, we are interested in estimating the moment parameters of a high-dimensional elliptically-contoured distribution. Existing estimators in low-dimensional settings require plugging in an estimated precision matrix. Such estimators work unsatisfactorily in high-dimensional settings, as estimating a high-dimensional precision matrix is by itself a difficult task. We discover that moment estimation for elliptical distributions does not necessarily need knowledge of the full precision matrix. We propose a marginal aggregation estimator (MAE), which only requires estimating the diagonal of the covariance matrix. Assuming mild sparsity on the covariance matrix, we show that MAE has the same asymptotic variance as an ideal estimator that knows the true precision matrix. We also extend MAE to a blockwise aggregation estimator (BAE) by accommodating estimates of diagonal blocks of the covariance matrix. BAE further relaxes the covariance sparsity requirement in MAE. The performances of MAE and BAE are evaluated in extensive simulations and an application to financial returns.