The covariance matrix, which should be estimated from the data, plays an important role in many multivariate procedures, and its positive definiteness (PDness) is essential for the validity of the procedures. Recently, many regularized estimators have been proposed and shown to be consistent in estimating the true matrix and its support under various structural assumptions on the true covariance matrix. However, they are often not PD. In this paper, we propose a simple modification to make a regularized covariance matrix be PD while preserving its support and the convergence rate. We focus on the matrix \(\ell_{\infty }\) norm error in covariance matrix estimation because it could allow us to bound the error in the downstream multivariate procedure relying on it. Our proposal in this paper is an extension of the fixed support positive-definite (FSPD) modification by Choi et al. (2019) from spectral and Frobenius norms to the matrix \(\ell_{\infty }\) norm. Like the original FSPD, we consider a convex combination between the initial estimator (the regularized covariance matrix without PDness) and a given form of the diagonal matrix minimize the \(\ell_{\infty }\) distance between the initial estimator and the convex combination, and find a closed-form expression for the modification. We apply the procedure to the minimum variance portfolio (MVP) optimization problem and show that the vector \(\ell_{\infty }\) error in the estimation of the optimal portfolio weight is bounded by the matrix \(\ell _{\infty }\) error of the plug-in covariance matrix estimator. We illustrate the MVP results with S&P 500 daily returns data from January 1978 to December 2014.

应从数据中估计出的协方差矩阵在许多多元过程中都起着重要作用，其正定性（PDness）对于过程的有效性至关重要。最近，已经提出了许多正则估计量，并证明它们在估计真实协方差矩阵的各种结构假设下在估计真实矩阵及其支持方面是一致的。但是，它们通常不是PD。在本文中，我们提出了一个简单的修改，以使正则化协方差矩阵成为PD，同时保留其支持和收敛速度。我们关注矩阵\（\ ell _ {\ infty} \）协方差矩阵估计中的范数误差，因为它可以使我们在依赖该误差的下游多变量过程中进行约束。我们在本文中提出的建议是Choi等人对固定支撑正定（FSPD）修改的扩展。（2019）从频谱和Frobenius规范到矩阵\（\ ell _ {\ infty} \）规范。像原始的FSPD一样，我们考虑初始估计量（无PDness的正则协方差矩阵）和对角矩阵的给定形式之间的凸组合最小化了初始估计量和凸量之间的\（\ ell _ {\ infty} \）距离组合，并找到用于修改的封闭形式的表达式。我们将该程序应用于最小方差投资组合（MVP）优化问题，并证明了向量\（\ ELL _ {\ infty} \）在最佳组合重量的估计误差是由矩阵界定\（\ ELL _ {\ infty} \）插件协方差矩阵估计的误差。我们用1978年1月至2014年12月的标准普尔500指数每日收益数据说明了MVP结果。