Abstract We provide an upper bound as a random variable for the functions of estimators in high dimensions. This upper bound may help establish the rate of convergence of functions in high dimensions. The upper bound random variable may converge faster, slower, or at the same rate as estimators depending on the behavior of the partial derivative of the function. We illustrate this via three examples. The first two examples use the upper bound for testing in high dimensions, and third example derives the estimated out-of-sample variance of large portfolios. All our results allow for a larger number of parameters, p, than the sample size, n.

中文翻译：

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高维估计器函数的上限

摘要 我们为高维估计器的函数提供了一个上限作为随机变量。这个上限可能有助于建立高维函数的收敛速度。取决于函数偏导数的行为，上限随机变量可能收敛得更快、更慢或与估计量相同。我们通过三个例子来说明这一点。前两个示例使用上限进行高维测试，第三个示例导出大型投资组合的估计样本外方差。我们所有的结果都允许使用比样本量 n 更大的参数 p。