In this paper, we analyze the statistic which is the difference in the values of an estimating function evaluated at its local maxima on two different subsets of the parameter space, assuming that the true parameter is in each subset, but possibly on the boundary. Our results extend known methods by covering a large class of estimation problems which allow sampling from nonidentically distributed random variables. Specifically, the existence and consistency of the local maximum estimators and asymptotic properties of useful hypothesis tests are obtained under certain law of large number and central limit-type assumptions. Other models covered include those with general log-likelihoods and/or covariates. As an example, the large sample theory of two-way nested random variance components models with covariates is derived from our main results.
"Generalization of likelihood ratio tests under nonstandard conditions." Ann. Statist. 25 (2) 897 - 916, April 1997. https://doi.org/10.1214/aos/1031833677