  We propose a "small-uniform" statistic for the inference of the functional
PCA estimator in a functional linear regression model. The literature has shown
two extreme behaviors: on the one hand, the FPCA estimator does not converge in
distribution in its norm topology; but on the other hand, the FPCA estimator
does have a pointwise asymptotic normal distribution. Our statistic takes a
middle ground between these two extremes: after a suitable rate normalization,
our small-uniform statistic is constructed as the maximizer of a fractional
programming problem of the FPCA estimator over a finite-dimensional subspace,
and whose dimensions will grow with sample size. We show the rate for which our
scalar statistic converges in probability to the supremum of a Gaussian
process. The small-uniform statistic has applications in hypothesis testing.
Simulations show our statistic has comparable to slightly better power
properties for hypothesis testing than the two statistics of Cardot, Ferraty,
Mas and Sarda (2003).
