Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the power of Sobolev tests for isotropy under local rotationally
  symmetric alternatives   We consider one of the most classical problems in multivariate statistics,
namely the problem of testing isotropy, or equivalently, the problem of testing
uniformity on the unit hypersphere $\mathcal{S}^{p-1}$ of $\mathbb{R}^p$.
Rather than restricting to tests that can detect specific types of alternatives
only, we consider the broad class of Sobolev tests. While these tests are known
to allow for omnibus testing of uniformity, their non-null behavior and
consistency rates, unexpectedly, remain largely unexplored. To improve on this,
we thoroughly study the local asymptotic powers of Sobolev tests under the most
classical alternatives to uniformity, namely, under rotationally symmetric
alternatives. We show in particular that the consistency rate of Sobolev tests
does not only depend on the coefficients defining these tests but also on the
derivatives of the underlying angular function at zero. For any Sobolev test
and any rotationally symmetric alternative, we derive the consistency rate of
the Sobolev test and determine the corresponding local asymptotic powers. We
show that Sobolev tests with non-zero coefficients at odd (respectively, even)
ranks only are blind (at any polynomial rate) to alternatives with angular
functions whose $k$th-order derivatives at zero vanish for any $k$ odd
(respectively, even). Our asymptotic analysis requires investigating the
non-standard behavior of random Chebyshev polynomials (for $p=2$) and random
Gegenbauer polynomials (for $p\geq 3$) in the vicinity of the uniform
distribution on $\mathcal{S}^{p-1}$. Our non-standard asymptotic results are
illustrated with Monte Carlo exercises.

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The categories of this paper are: bayes-an
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The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-th
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: eess
0
The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph