A geometric approach to confidence regions and bands for functional parameters

Hyunphil Choi, Matthew Reimherr

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

Functional data analysis is now a well-established discipline of statistics, with its core concepts and perspectives in place. Despite this, there are still fundamental statistical questions which have received relatively little attention. One of these is the systematic construction of confidence regions for functional parameters. This work is concerned with developing, understanding and visualizing such regions. We provide a general strategy for constructing confidence regions in a real separable Hilbert space by using hyperellipsoids and hyper-rectangles. We then propose specific implementations which work especially well in practice. They provide powerful hypothesis tests and useful visualization tools without relying on simulation. We also demonstrate the negative result that nearly all regions, including our own, have zero coverage when working with empirical covariances. To overcome this challenge we propose a new paradigm for evaluating confidence regions by showing that the distance between an estimated region and the desired region (with proper coverage) tends to 0 faster than the regions shrink to a point. We call this phenomena ghosting and refer to the empirical regions as ghost regions. We illustrate the proposed methods in a simulation study and an application to fractional anisotropy tract profile data.

Original languageEnglish (US)
Pages (from-to)239-260
Number of pages22
JournalJournal of the Royal Statistical Society. Series B: Statistical Methodology
Volume80
Issue number1
DOIs
StatePublished - Jan 2018

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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