Corpus ID: 230433807

Statistical Inference on the Hilbert Sphere with Application to Random Densities

@inproceedings{Dai2021StatisticalIO,
  title={Statistical Inference on the Hilbert Sphere with Application to Random Densities},
  author={Xiongtao Dai},
  year={2021}
}
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… Expand

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References

SHOWING 1-10 OF 57 REFERENCES
Fréchet analysis of variance for random objects
Fréchet mean and variance provide a way of obtaining a mean and variance for metric space-valued random variables, and can be used for statistical analysis of data objects that lie in abstractExpand
Large sample theory of intrinsic and extrinsic sample means on manifolds--II
This article develops nonparametric inference procedures for estimation and testing problems for means on manifolds. A central limit theorem for Frechet sample means is derived leading to anExpand
A smeary central limit theorem for manifolds with application to high-dimensional spheres
The (CLT) central limit theorems for generalized Frechet means (data descriptors assuming values in stratified spaces, such as intrinsic means, geodesics, etc.) on manifolds from the literature areExpand
LARGE SAMPLE THEORY OF INTRINSIC AND EXTRINSIC SAMPLE MEANS ON MANIFOLDS—II
This article develops nonparametric inference procedures for estimation and testing problems for means on manifolds. A central limit theorem for Frechet sample means is derived leading to anExpand
Nonparametric estimation of means on Hilbert manifolds and extrinsic analysis of mean shapes of contours
TLDR
A general extrinsic approach for data analysis on Hilbert manifolds with a focus on means of probability distributions on such sample spaces is introduced, appealing to the concept of neighborhood hypotheses from functional data analysis and derive a one-sample test. Expand
Functional data analysis for density functions by transformation to a Hilbert space
Functional data that are nonnegative and have a constrained integral can be considered as samples of one-dimensional density functions. Such data are ubiquitous. Due to the inherent constraints,Expand
Principal component analysis for functional data on Riemannian manifolds and spheres
Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered for example as movement trajectories on the surface of the earth, areExpand
On the meaning of mean shape: manifold stability, locus and the two sample test
Various concepts of mean shape previously unrelated in the literature are brought into relation. In particular, for non-manifolds, such as Kendall’s 3D shape space, this paper answers the question,Expand
Intrinsic means on the circle: uniqueness, locus and asymptotics
This paper gives a comprehensive treatment of local uniqueness, asymptotics and numerics for intrinsic sample means on the circle. It turns out that local uniqueness as well as rates of convergenceExpand
Functional models for time-varying random objects
In recent years, samples of time-varying object data such as time-varying networks that are not in a vector space have been increasingly collected. These data can be viewed as elements of a generalExpand
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