椭圆分布数据的高阶矩估计：有必要用大锤敲碎鸡蛋吗？
Higher Moment Estimation for Elliptically-distributed Data: Is it Necessary to Use a Sledgehammer to Crack an Egg?
摘要：
多元椭圆等高分布广泛用于经济和金融数据建模。研究了高维情形下半参数椭圆模型矩参数的估计问题。这类估计对于金融数据分析和二次判别分析是有用的。对于低维椭圆模型，有效的矩估计可以通过插入精度矩阵的估计来获得。插件估计器到高维设置的自然推广由于估计大的精度矩阵而执行得不令人满意。我们真的需要一把大锤来打破鸡蛋吗？幸运的是，我们发现矩参数可以有效地估计，而无需估计高维精度矩阵。提出了矩参数的边际聚集估计（MAE）.该算法只需要估计协方差阵的对角线，便于实现.在协方差结构具有弱稀疏性的情况下，证明了MAE的渐近方差与已知真精度矩阵的理想插入估计量的渐近方差相同，因此MAE是渐近有效的。当协方差阵的对角块的估计可用时，我们还将MAE扩展到块聚集估计（BAE）。我们的方法的性能进行了验证，通过广泛的模拟和应用程序的财务回报。
英文摘要：
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.
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