A Fast Algorithm for Log-Concave Density Estimation

• Activity Number: 74
• Type: Contributed
• Date/Time: Sunday, August 9, 2015 : 4:00 PM to 5:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Yu Liu* and Yong Wang
• Companies: and The University of Auckland
• Keywords: Log-concave ; Density estimation ; Nonparametric maximum likelihood ; Quadratic approximation

Abstract

A new fast algorithm is proposed and studied for computing the nonparametric maximum likelihood estimate of a log-concave density. It is an extension of the constrained Newton method that was proposed for nonparametric mixture estimation. In each iteration, the new algorithm includes, if necessary, new knots in aid of a gradient function, renews the changes of slope at all knots via a quadratically convergent method and removes the knots at which the changes of slope become zero. The new algorithm is theoretically guaranteed to converge to the unique maximum likelihood estimate. As shown in numerical studies, it outperforms algorithms available in the literature. Its applications to some real-world financial data are also given.