  Profile likelihood confidence intervals are a robust alternative to Wald's
method if the asymptotic properties of the maximum likelihood estimator are not
met. However, the constrained optimization problem defining profile likelihood
confidence intervals can be difficult to solve in these situations, because the
likelihood function may exhibit unfavorable properties. As a result, existing
methods may be inefficient and yield misleading results. In this paper, we
address this problem by computing profile likelihood confidence intervals via a
trust-region approach, where steps computed based on local approximations are
constrained to regions where these approximations are sufficiently precise. As
our algorithm also accounts for numerical issues arising if the likelihood
function is strongly non-linear or parameters are not estimable, the method is
applicable in many scenarios where earlier approaches are shown to be
unreliable. To demonstrate its potential in applications, we apply our
algorithm to benchmark problems and compare it with 6 existing approaches to
compute profile likelihood confidence intervals. Our algorithm consistently
achieved higher success rates than any competitor while also being among the
quickest methods. As our algorithm can be applied to compute both confidence
intervals of parameters and model predictions, it is useful in a wide range of
scenarios.
