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We propose confidence regions with asymptotically correct uniform coverageprobability of parameters whose Fisher information matrix can be singular atimportant points of the parameter set. Our work is motivated by the need forreliable inference on scale parameters close or equal to zero in mixed models,which is obtained as a special case. The confidence regions are constructed byinverting a continuous extension of the score test statistic standardized byexpected information, which we show exists at points of singular informationunder regularity conditions. Similar results have previously only been obtainedfor scalar parameters, under conditions stronger than ours, and applications tomixed models have not been considered. In simulations our confidence regionshave near-nominal coverage with as few as $n = 20$ independent observations,regardless of how close to the boundary the true parameter is. It is acorollary of our main results that the proposed test statistic has anasymptotic chi-square distribution with degrees of freedom equal to the numberof tested parameters, even if they are on the boundary of the parameter set.

Confidence Regions Near Singular Information and Boundary Points With Applications to Mixed Models