英文标题：
《Optimal Linear Shrinkage Estimator for Large Dimensional Precision
  Matrix》
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作者：
Taras Bodnar, Arjun K. Gupta and Nestor Parolya
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最新提交年份：
2014
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英文摘要：
  In this work we construct an optimal shrinkage estimator for the precision matrix in high dimensions. We consider the general asymptotics when the number of variables $p\\rightarrow\\infty$ and the sample size $n\\rightarrow\\infty$ so that $p/n\\rightarrow c\\in (0, +\\infty)$. The precision matrix is estimated directly, without inverting the corresponding estimator for the covariance matrix. The recent results from the random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The resulting distribution-free estimator has almost surely the minimum Frobenius loss. Additionally, we prove that the Frobenius norms of the inverse and of the pseudo-inverse sample covariance matrices tend almost surely to deterministic quantities and estimate them consistently. At the end, a simulation is provided where the suggested estimator is compared with the estimators for the precision matrix proposed in the literature. The optimal shrinkage estimator shows significant improvement and robustness even for non-normally distributed data.
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中文摘要：
本文构造了高维精度矩阵的最优收缩估计。当变量的数量为$p\\rightarrow\\infty$和样本量为$n\\rightarrow\\infty$时，我们考虑一般渐近性，使得$p/n\\rightarrow c\\在（0，+\\infty）$。直接估计精度矩阵，而不反转协方差矩阵的相应估计量。随机矩阵理论的最新结果允许我们找到最佳收缩强度的渐近确定性等价物，并一致地估计它们。由此得到的无分布估计几乎肯定具有最小的Frobenius损失。此外，我们还证明了逆样本协方差矩阵和伪逆样本协方差矩阵的Frobenius范数几乎肯定倾向于确定性量，并一致地估计了它们。最后，给出了一个仿真结果，并与文献中提出的精度矩阵的估计值进行了比较。即使对于非正态分布的数据，最优收缩估计也显示出显著的改进和鲁棒性。
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分类信息：

一级分类：Mathematics        数学
二级分类：Statistics Theory        统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance        数量金融学
二级分类：Statistical Finance        统计金融
分类描述：Statistical, econometric and econophysics analyses with applications to financial markets and economic data
统计、计量经济学和经济物理学分析及其在金融市场和经济数据中的应用
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一级分类：Statistics        统计学
二级分类：Statistics Theory        统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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