Optimal and Adaptive Estimation of Extreme Values in the Permuted Monotone Matrix Model

• Activity Number: 174 - Statistical Optimality in High-Dimensional Models and Tradeoffs with Computational Complexity, Privacy and Communication Constraints
• Type: Contributed
• Date/Time: Tuesday, August 4, 2020 : 10:00 AM to 2:00 PM
• Sponsor: IMS
• Author(s): Rong Ma* and Tony Cai and Hongzhe Li
• Companies: University of Pennsylvania and University of Pennsylvania and University of Pennsylvania
• Keywords: compound decision problem; extreme values; minimax lower bounds; monotone matrix; permutation

Abstract

Motivated by applications in metagenomics, we consider the permuted monotone matrix model where the observed data matrix is the sum of a column-permuted signal matrix with monotone rows and a noise matrix. This paper studies the estimation of the extreme values associated to the signal matrix, including its first and last columns, as well as their difference (the range vector). Treating these estimation problems as compound decision problems, minimax rate-optimal and adaptive estimators are constructed using the spectral column sorting method. Novel techniques that can be effective in estimating an arbitrary high-dimensional nonlinear operator are developed to establish minimax lower bounds, including generalized Le Cam's method and Fano's method. Numerical experiments using simulated and synthetic microbiome metagenomic data are presented, showing the superiority of the proposed methods over the alternatives. The methods are illustrated by comparing the growth rates of gut bacteria between inflammatory bowel disease patients and normal controls.