Asymptotics for likelihood ratio tests of boundary points with singular information and unidentifiable nuisance parameters

We establish the asymptotic distribution of likelihood ratio tests (LRTs) in settings where some of the nuisance parameters are unidentifiable under the null hypothesis, parameters of interest lie on the boundary of the parameter space, and the information matrix of the identifiable parameters may be singular. Our work is motivated by mixture models and genetic linkage analysis, which exhibit all three features simultaneously, but it is applicable more broadly to other problems such as change-point detection. Under suitable regularity conditions, the asymptotic distribution of the LRT statistic under the null hypothesis is the supremum of a $\bar{\chi}^2$-process, that is, a stochastic process whose marginal distributions are mixtures of $\chi^2$-distributions with weights depending on the nuisance parameter. Under local alternatives, the asymptotic distribution of the LRT statistic is the supremum of a noncentral $\bar{\chi}^2$-process, whose marginal distributions are mixtures of truncated, noncentral $\chi^2$-distributions. In contrast to prior work on singular information, where singularity stems from the parameter of interest and changes the form of the limit distribution, here singularity is determined by the nuisance parameter and the limit has the same form as in the nonsingular case. Existing results for boundary inference with nonsingular information or without nuisance parameters are obtained as special cases, and several existing application-specific results for mixture models and genetic linkage analysis are recovered and extended.