Plotting Likelihood-Ratio-Based Confidence Regions for Two-Parameter Univariate Probability Models

Abstract

Plotting two-parameter confidence regions is nontrivial. Numerical methods often rely on a computationally expensive grid-like exploration of the parameter space. A recent advance reduces the two-dimensional problem to many one-dimensional problems employing a trigonometric transformation that assigns an angle $\varphi$ from the maximum likelihood estimator, and an unknown radial distance to its confidence region boundary. This paradigm shift can improve computational runtime by orders of magnitude, but it is not robust. Specifically, parameters differing greatly in magnitude and/or challenging nonconvex confidence region shapes make the plot susceptible to inefficiencies and/or inaccuracies. This article improves the technique by (i) keeping confidence region boundary searches in the parameter space, (ii) selectively targeting confidence region boundary points in lieu of uniformly spaced $\varphi$ angles from the maximum likelihood estimator and (iii) enabling access to regions otherwise unreachable due to multiple roots for select $\varphi$ angles. Two heuristics are given for $\varphi$ selection: an elliptic-inspired angle selection heuristic and an intelligent smoothing search heuristic. Finally, a jump-center heuristic permits plotting otherwise inaccessible multiroot regions. This article develops these heuristics for two-parameter likelihood-ratio-based confidence regions associated with univariate probability distributions, and introduces the R conf package, which automates the process and is publicly available via CRAN.