New experimental data on the isotopic variations of neon, argon, and xenon in a popping rock imply that the 40Ar/36Ar ratio of the upper mantle is less than 44,000 and that the 129Xe/130Xe ratio is less than 8.2. The elemental abundance pattern of rare gases is chondritic-like and is quite distinct from the solar pattern. These data imply that Earth accreted from planetesimals that probably underwent a transformation of their rare gas budget from solar- to chondritic-like, leaving the isotopic composition unchanged from the solar pattern.
This paper studies the asymptotic power of tests of sphericity against
perturbations in a single unknown direction as both the dimensionality of the
data and the number of observations go to infinity. We establish the
convergence, under the null hypothesis and contiguous alternatives, of the log
ratio of the joint densities of the sample covariance eigenvalues to a Gaussian
process indexed by the norm of the perturbation. When the perturbation norm is
larger than the phase transition threshold studied in Baik, Ben Arous and Peche
[Ann. Probab. 33 (2005) 1643-1697] the limiting process is degenerate, and
discrimination between the null and the alternative is asymptotically certain.
When the norm is below the threshold, the limiting process is nondegenerate,
and the joint eigenvalue densities under the null and alternative hypotheses
are mutually contiguous. Using the asymptotic theory of statistical
experiments, we obtain asymptotic power envelopes and derive the asymptotic
power for various sphericity tests in the contiguity region. In particular, we
show that the asymptotic power of the Tracy-Widom-type tests is trivial (i.e.,
equals the asymptotic size), whereas that of the eigenvalue-based likelihood
ratio test is strictly larger than the size, and close to the power envelope.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1100 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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