A geometric approach to confidence regions and bands for functional parameters

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2018 · Volume 80 · Issue 1

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, havezero coveragewhen 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 phenomenaghostingand refer to the empirical regions asghostregions. We illustrate the proposed methods in a simulation study and an application to fractional anisotropy tract profile data.

Keywords

Confidence bands, Confidence regions, Functional data analysis, Hypothesis testing, Principal component analysis

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