  We show that confidence intervals for a variance component or proportion,
with asymptotically correct uniform coverage probability, can be obtained by
inverting certain test-statistics based on the score for the restricted
likelihood. The results apply in settings where the variance or proportion is
near or at the boundary of the parameter set. Simulations indicate the proposed
test-statistics are approximately pivotal and lead to confidence intervals with
near-nominal coverage even in small samples. We illustrate our methods'
application in spatially-resolved transcriptomics where we compute
approximately 15,000 confidence intervals, used for gene ranking, in less than
4 minutes. In the settings we consider, the proposed method is between two and
28,000 times faster than popular alternatives, depending on how many confidence
intervals are computed.
