The LRT statistic converges in distribution to the supremum of a bar-chi-squared process under the null and a noncentral version under local alternatives, with the same form whether or not the information matrix is singular due to the nuisance parameter.
Confidence regions near singular information and boundary points with applications to mixed models
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Asymptotics for likelihood ratio tests of boundary points with singular information and unidentifiable nuisance parameters
The LRT statistic converges in distribution to the supremum of a bar-chi-squared process under the null and a noncentral version under local alternatives, with the same form whether or not the information matrix is singular due to the nuisance parameter.