The main contribution of this paper is the derivation of the asymptotic behavior of the out-of-sample variance, the out-of-sample relative loss, and of their empirical counterparts in the high-dimensional setting, i.e., when both ratios $p/n$ and $p/m$ tend to some positive constants as $m\to \infty$ and $n\to \infty$, where $p$ is the portfolio dimension, while $n$ and $m$ are the sample sizes from the in-sample and out-of-sample periods, respectively. The results are obtained for the traditional estimator of the global minimum variance (GMV) portfolio and for the two shrinkage estimators introduced by Frahm and Memmel (2010) and Bodnar et al. (2018). We show that the behavior of the empirical out-of-sample variance may be misleading in many practical situations, leading, for example, to a comparison of zeros. On the other hand, this will never happen with the empirical out-of-sample relative loss, which seems to provide a natural normalization of the out-of-sample variance in the high-dimensional setup. As a result, an important question arises if the out-of-sample variance can safely be used in practice for portfolios constructed from a large asset universe.

本文的主要贡献是推导了样本外方差的渐近行为、样本外相对损失以及它们在高维设置中的经验对应物，即当两个比率$p/n$和$p/米$倾向于一些正常数作为$米\to \infty$和$n\to \infty$， 在哪里$p$是投资组合维度，而$n$和$米$分别是样本内和样本外时期的样本量。结果是针对全局最小方差 (GMV) 投资组合的传统估计器以及 Frahm 和 Memmel（2010 年）以及 Bodnar 等人引入的两个收缩估计器获得的。(2018)。我们表明，经验样本外方差的行为在许多实际情况下可能会产生误导，例如，导致对零的比较。另一方面，经验样本外相对损失永远不会发生这种情况，这似乎提供了高维设置中样本外方差的自然归一化。因此，一个重要的问题出现了，即样本外方差是否可以在实践中安全地用于从大型资产领域构建的投资组合。