Active set algorithms for estimating shape-constrained density ratios

We develop an active set algorithm for the nonparametric estimation of log-concave or log-convex density ratios. In the setting of log-concave densities, our algorithm is similar to, but faster than the one of Duembgen et al. (2011). Our general framework includes also the estimation of a log-convex tail inflation function as proposed by McCullagh and Polson (2012). As a by-product, we obtain new goodness-of-fit tests of single hypotheses which are often more powerful than higher criticism tests.