Statistical Inference on the Hilbert Sphere with Application to Random Densities

Abstract

The infinite-dimensional Hilbert sphere S ∞ has been widely employed to model density functions and shapes, extending the finite-dimensional counterpart. We consider the Fréchet mean as an intrinsic summary of the central tendency of data lying on S ∞ . To break a path for sound statistical inference, we derive properties of the Fréchet mean on S ∞ 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 ∞ . Intrinsic CLTs for the estimated tangent vectors and covariance operator are also obtained. Asymptotic and bootstrap hypothesis tests for the Fréchet 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.

Intrinsic Wasserstein Correlation Analysis

“…In order to tackle the nonlinear structure of the Wasserstein space, Petersen et al (2016) proposed a log quantile density (LQD) transformation to turn probability density functions to unconstrained functions. Dai (2021) adopted a square root transformation to map density functions into the positive orthant of a unit Hilbert sphere S ∞ . However, none of these consider the more natural geometry that is compatible with optimal transport on the Wasserstein space.…”