Density Estimation in the Two-Sample Problem with Likelihood Ratio Ordering

• Activity Number: 18
• Type: Topic Contributed
• Date/Time: Sunday, August 9, 2015 : 2:00 PM to 3:50 PM
• Sponsor: International Chinese Statistical Association
• Author(s): Tao Yu* and Pengfei Li and Jing Qin
• Companies: National University of Singapore and University of Waterloo and National Institute of Allergy and Infectious Diseases
• Keywords: Contractivity ; Greatest convex minorant ; Likelihood ratio ordering ; Pool-adjacent-violators algorithm ; Smoothed likelihood ; Weighted isotonic regression

Abstract

In this paper, we propose a method for estimating the probability density functions in a two sample problem where the ratio of the densities is monotonic. Such a problem is well motivated from medical data and has been widely identified in the literature. However, effective methods for solving this problem are not yet available in the community. Clearly, an effectively method should at least satisfy (1) the resulting estimates are probability densities; (2) the corresponding density ratio inherits the monotonic property, otherwise it is difficult to explain the analysis results. In this paper, we propose estimates for these densities whose ratio inherits the monotonic property, and explore their theoretical properties. One interesting application of our density estimates is that the corresponding receiver operating characteristic (ROC) curve estimate is concave. Through numerical studies, we observe that both the density estimates and the ROC curve estimate from our method outperform their competitors, particularly when the sample size is relatively small.