Summary This paper is concerned with empirical likelihood inference on the population mean when the dimension $p$ and the sample size $n$ satisfy $pbecomes too small to cover the true mean value. Moreover, when $p> n$, the sample covariance matrix becomes singular, and this results in the breakdown of the first sandwich approximation for the log empirical likelihood ratio. To deal with these two challenges, we propose a n/n\rightarrow c\in [1,\infty)$. As shown in Tsao (2004), the empirical likelihood method fails with high probability when $p/n>1/2$ because the convex hull of the $n$ observations in $\mathbb{R}^p$ ew strategy of adding two artificial data points to the observed data. We establish the asymptotic normality of the proposed empirical likelihood ratio test. The proposed test statistic does not involve the inverse of the sample covariance matrix. Furthermore, its form is explicit, so the test can easily be carried out with low computational cost. Our numerical comparison shows that the proposed test outperforms some existing tests for high-dimensional mean vectors in terms of power. We also illustrate the proposed procedure with an empirical analysis of stock data.

中文翻译：

---

大维均值向量的经验似然检验

总结 本文关注的是当维度 $p$ 和样本大小 $n$ 满足 $p 变得太小而无法覆盖真实平均值时，对总体均值的经验似然推断。此外，当 $p>n$ 时，样本协方差矩阵变得奇异，这导致对数经验似然比的第一三明治近似的分解。为了应对这两个挑战，我们提出了一个/n\rightarrow c\in [1,\infty)$。如 Tsao (2004) 所示，当 $p/n>1/2$ 时，经验似然方法失败的概率很高，因为 $\mathbb{R}^p$ 中的 $n$ 观测值的凸包两个人工数据指向观察到的数据。我们建立了所提出的经验似然比检验的渐近正态性。建议的检验统计量不涉及样本协方差矩阵的逆。此外，它的形式是明确的，因此可以轻松地以较低的计算成本进行测试。我们的数值比较表明，所提出的测试在功效方面优于一些现有的高维平均向量测试。我们还通过对股票数据的实证分析来说明所提出的程序。