The null distribution of distance covariance is usually approximated by permutation, which is prohibitive when very small p-values are needed, or by matching a few moments to a parametric family, which is inaccurate in the tails. A third option is to approximate the limiting distribution, a weighted sum of chi-square variables, directly through the spectra of the doubly centred distance matrices. This is used for kernel-based tests but has lacked a rigorous justification. We prove that the empirical spectra give a uniformly consistent approximation of the limiting null distribution, and hence an asymptotically valid test, for a general class of distances of negative type on separable metric spaces. The result covers the Hilbert-Schmidt independence criterion as a special case. We also give an adaptive algorithm that brackets the p-value from a partial eigendecomposition, reducing the cost from $O(n^3)$ to $O(k n^2)$, and a shrinkage correction matching the first two moments. In simulations, the proposed tests are the only non-Monte-Carlo procedures whose empirical type I error converges to the nominal level.
The Distance Profile Embedding is introduced, a novel representation that maps random objects from general metric spaces into a Hilbert space of square-integrable functions and it is proved that this mapping is injective and preserves full distributional information without requiring isometric Hilbert embeddings or one...
The class of parametric statistical models that can be treated as Riemannian manifolds is considerably larger than the classical Fisher-Rao setting allows, once one works in the space of tempered distributions. A law is represented by a tempered distribution T in S'(R^k), while an instrument - a positive Schwartz kerne...
Comparing correlation matrices across time or stress scenarios is critical in quantitative finance and multivariate statistics, yet sample estimation noise often obscures whether an observed distance reflects a true structural shift. We derive the asymptotic sampling distribution of the intrinsic off-log (log-Euclidean...
We study asymptotic properties of statistics in goodness-of-fit tests under the simple null hypothesis of a multinomial distribution. It is also assumed that the statistics are based on $\phi$-divergences, the sample size $n$ tends to infinity, and the number of cells in the group $k$ is fixed. It is known that in this...
V. Ulyanov, A. V. Sheryaev· Theory of Probability &...· 0 citations
Correlation matrices arise when marginal scales are removed from covariance matrices, yet a normalized likelihood must account for both quotient distance and quotient volume. We propose a Riemannian Gaussian model for full-rank correlation matrices under quotient-affine geometry. The distribution is proper and has fini...
Maximum-entropy reference distributions are usually constructed on the normalized probability simplex. This formulation is less natural for unnormalized statistical models, in which positive multiples represent the same shape, and it does not directly explain how a prescribed admissible region should determine the defo...
H. Hino· 1 citation
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