Statistical Inference on the Hilbert Sphere with Application to Random Densities

TL;DR

This paper develops statistical inference methods for data on the infinite-dimensional Hilbert sphere, focusing on the Fréchet mean, CLTs, and hypothesis tests, with applications to density functions and shapes.

Contribution

It establishes the existence, uniqueness, and asymptotic distribution of the Fréchet mean on the Hilbert sphere, enabling sound statistical inference in infinite-dimensional settings.

Findings

Proved root-n CLT for the Fréchet mean on $S^ Infty$

Developed consistent hypothesis tests based on intrinsic geometry

Demonstrated the effectiveness of the methods on density data from real-world applications

Abstract

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.