Shape-Constrained Inference for Concave-Transformed Densities and their Modes

Abstract

We consider inference about functions estimated via shape constraints based on concavity. We consider log-concave densities and other “concave-transformed” densities on the real line, where a concave-transformed class is one given by applying a transformation (e.g. the logarithm or a power function) to concave functions. We expect our proofs and results to be relevant in other concavity-based settings. Concave functions are always unimodal, so concave-transformed densities can be used as surrogates for unimodal ones, and the mode is thus a natural parameter of interest. In nonparametric settings the mode is generally not estimable at a root-n rate and does not always have a normal limiting distribution, and current methods for testing or forming confidence intervals for the location of the mode are generally complicated. In the setting of log-concave density estimation we construct a likelihood ratio test for the location of the mode by comparing the log-concave maximum likelihood estimate (MLE) to the MLE over the constrained subclass of log-concave densities with a fixed mode. The test can be inverted to form a confidence set. We study the properties of the constrained MLE and the Wilks phenomenon of the likelihood ratio statistic. Proving global rates of convergence of n^{2/5}, for both the constrained and unconstrained MLEs, is an important step in understanding the likelihood ratio statistic and this result is also of independent interest. These global rate results apply to Hellinger and total variation distance, as well as to the size of the likelihood ratio statistic, and they apply to many concave-transformed density classes beyond log-concave ones.