  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.
