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