On the Minimax Risks for Estimation of a Location Parameter from Sums

• Activity Number: 107
• Type: Contributed
• Date/Time: Monday, July 30, 2007 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Mokshay Madiman*+ and Andrew Barron and Abram Kagan and Tinghui Yu
• Companies: Yale University and Yale University and University of Maryland and University of Maryland
• Keywords: Minimax risk ; Pitman estimator ; Distributed estimation ; Fisher information ; Information inequalities

Abstract

Suppose there are N random variables of interest, and each user has access to observations from the location family generated by a particular subset sum. For a given class of subsets, the variances of the Pitman estimates obtained by the corresponding class of users is related to the variance of the Pitman estimate obtained by a user who only sees observations from the location family generated by the total sum. This relationship is made explicit through a new inequality, which strengthens various inequalities for the Fisher information of a sum due to Stam (1959), Artstein, Ball, Barthe and Naor (2004), and Madiman and Barron (2006). In addition to providing insights on distributed estimation, the new result relates the minimax risks (in terms of mean squared error) incurred for estimating a location parameter by users who have access to different sets of observations.