In many instances, imposing a constraint on the shape of a density is a reasonable and flexible assumption. It offers an alternative to parametric models, which can be too rigid, and to other nonparametric methods, which require the choice of tuning parameters. The nonparametric estimation of log-concave or log-convex density ratios is treated by means of active set algorithms in a unified framework. In the setting of log-concave densities, the new algorithm is similar to, but substantially faster than, previously considered active set methods. Log-convexity, on the other hand, is a less common shape-constraint, described by some authors as “tail inflation”. The active set method proposed here is novel in this context. As a by-product, new goodness-of-fit tests of single hypotheses are formulated and are shown to be more powerful than higher criticism tests in a simulation study.
