Likelihood-Based Confidence Curves for Bounded Parameters
When a parameter is bounded, the likelihood ratio statistic does not generally follow the chi-square limit distribution of Wilks' theorem. In fact, its asymptotic null distribution depends on the distance from the null value to the boundary, interpolating between the chi-bar-square law on the boundary and the chi-square law in the deep interior of the parameter space. We derive this distribution and use it to construct confidence curves, which are nested collections of two-sided confidence intervals or regions, for bounded parameters. We extend the existing result for the scalar parameter to a two-dimensional parameter with one or both components bounded. As the bivariate distributions are integrals of normal distributions, which are not reducible to standard statistical functions, we also give a simpler approximated distribution obtained by moment-matching. We illustrate the methods on a canonical analysis from high-energy physics, namely the Poisson signal-plus-background experiment.