  The infinite-dimensional Hilbert sphere $S^\infty$ has been widely employed
to model density functions and shapes, extending the finite-dimensional
counterpart. We consider the Fr\'echet mean as an intrinsic summary of the
central tendency of data lying on $S^\infty$. To break a path for sound
statistical inference, we derive properties of the Fr\'echet mean on $S^\infty$
by establishing its existence and uniqueness as well as a root-$n$ central
limit theorem (CLT) for the sample version, overcoming obstructions from
infinite-dimensionality and lack of compactness on $S^\infty$. Intrinsic CLTs
for the estimated tangent vectors and covariance operator are also obtained.
Asymptotic and bootstrap hypothesis tests for the Fr\'echet mean based on
projection and norm are then proposed and are shown to be consistent. The
proposed two-sample tests are applied to make inference for daily taxi demand
patterns over Manhattan modeled as densities, of which the square roots are
analyzed on the Hilbert sphere. Numerical properties of the proposed hypothesis
tests which utilize the spherical geometry are studied in the real data
application and simulations, where we demonstrate that the tests based on the
intrinsic geometry compare favorably to those based on an extrinsic or flat
geometry.
