Likelihood ratio processes under nonstandard settings

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.