Statistical inference on the Hilbert sphere with application to random densities

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