Theory of Probability &Its Applications, Volume 67, Issue 2, Page 246-260, August 2022.
This paper establishes the LAN property for the curved normal families and the simultaneous equation systems. In addition, we show that one-way random ANOVA models fail to have the LAN property. We consider the two cases when the variance of random effect lies in the interior and boundary of parameter space. In the former case, the log-likelihood ratio converges to 0. In the latter case, the log-likelihood ratio has atypical limit distributions, which depend on the contiguity orders. The contiguity orders corresponding to the variances of random effects and disturbances can be equal to or greater than one, respectively, and that corresponding to the grand mean can be equal to or greater than one half. Consequently, we cannot use the ordinary optimal theory based on the LAN property. Meanwhile, the test based on the log-likelihood ratio is shown to be asymptotically most powerful with the benefit of the classical Neymann--Pearson framework.

中文翻译：

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非标准设置下的似然比过程

概率论及其应用，第 67 卷，第 2 期，第 246-260 页，2022 年 8 月。
本文建立了曲线正规族和联立方程组的 LAN 属性。此外，我们表明单向随机方差分析模型不具有 LAN 属性。我们考虑随机效应方差位于参数空间内部和边界的两种情况。在前一种情况下，对数似然比收敛到 0。在后一种情况下，对数似然比具有非典型的极限分布，这取决于邻接顺序。与随机效应和干扰的方差对应的邻接阶数可以分别等于或大于一，与全均值对应的邻接阶数可以等于或大于二分之一。因此，我们不能使用基于 LAN 特性的普通最优理论。同时，