Active set algorithms for estimating shape-constrained density ratios

TL;DR

This paper introduces efficient active set algorithms for nonparametric estimation of shape-constrained density ratios, specifically log-concave and log-convex, providing faster computation and new hypothesis testing methods.

Contribution

It presents a unified framework for estimating shape-constrained density ratios using active set algorithms, including a novel approach for log-convex densities and new goodness-of-fit tests.

Findings

The new algorithm for log-concave densities is significantly faster than previous methods.

Proposed tests outperform higher criticism tests in simulations.

Active set methods effectively estimate shape-constrained density ratios.

Abstract

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 requiring the choice of tuning parameters. This paper treats the nonparametric estimation of log-concave or log-convex density ratios 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 is a less common shape constraint which is 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.