A Geometric Approach to Confidence Regions and Bands for Functional Parameters

TL;DR

This paper develops a geometric framework for constructing and visualizing confidence regions for functional parameters in FDA, addressing coverage issues and introducing the concept of ghost regions for better evaluation.

Contribution

It introduces a general geometric strategy for confidence regions in Hilbert spaces, proposes practical implementations, and presents a new paradigm for assessing their accuracy.

Findings

Confidence regions often have zero coverage with empirical covariances.

Proposed methods effectively visualize and test hypotheses without simulation.

The ghosting paradigm improves evaluation of confidence regions.

Abstract

Functional data analysis, FDA, 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 using hyper-ellipsoids and hyper-rectangles. We then propose specific implementations which work especially well in practice. They provide powerful hypothesis tests and useful visualization tools without using any 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 zero 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.