Title: Likelihood ratio tests and confidence intervals based on the shape constraint of concavity
Authors: Charles Doss - University of Minnesota (United States) [presenting]
Jon A Wellner - University of Washington (United States)
Abstract: Estimation and inference for a log-concave density and for a concave regression function is considered. These problems have some similarities because they both rely on an underlying shape constraint of concavity. Forming confidence intervals or hypothesis tests in nonparametric settings is often challenging. We propose using likelihood ratio statistics to form hypothesis tests (which can be inverted to form confidence intervals). We consider tests or intervals for the location of the mode of the log-concave density function and for the value of the concave regression function. The statistics we propose are tuning parameter free, a rarity in nonparametric settings. We demonstrate that the likelihood ratio statistics are asymptotically pivotal (satisfy the so-called Wilks phenomenon). Thus, they have universal critical values not depending on any unknown parameters, allowing the tests or intervals to be computed in practice.