Let $x_1,\dots ,x_n$ be independent and identically distributed (i.i.d.) real-valued random vectors from distribution $N_p(\mu ,\mathrm{\Sigma })$, where the sample size $n$ and the vector dimension $p$ satisfy $n-1>p\to \infty$. We are interested in the exponential convergence rate of the likelihood ratio test (LRT) statistics for testing $\mathrm{\Sigma }$ equal to a given matrix and $(\mu ,\mathrm{\Sigma })$ equal to a given pair. In traditional statistical theory, the LRT statistics have been studied under the null hypothesis and finite-dimensional conditions. In this paper, we prove the moderate deviation principle (MDP) under the high-dimensional conditions for the two LRT statistics. We show that our results hold under the null hypothesis and the alternative hypothesis as well. Some numerical simulations indicate that our conclusions have good performance.

让$X_1,\dots \dots ,X_n$是来自分布的独立同分布 (iid) 实值随机向量${氮}_p（\mu ,\mathrm{\Sigma }）$，其中样本量$n$和向量维度$p$满足$n-1>p\to \mathrm{无穷大}$。我们感兴趣的是用于测试的似然比检验（LRT）统计的指数收敛速度$\mathrm{\Sigma }$等于给定矩阵并且$（\mu ,\mathrm{\Sigma }）$等于给定的对。在传统的统计理论中，LRT统计是在零假设和有限维条件下研究的。在本文中，我们证明了两个LRT统计量在高维条件下的适度偏差原理（MDP）。我们证明我们的结果在原假设和备择假设下也成立。一些数值模拟表明我们的结论具有良好的性能。