43,612 research outputs found
Strong approximations of level exceedences related to multiple hypothesis testing
Particularly in genomics, but also in other fields, it has become commonplace
to undertake highly multiple Student's -tests based on relatively small
sample sizes. The literature on this topic is continually expanding, but the
main approaches used to control the family-wise error rate and false discovery
rate are still based on the assumption that the tests are independent. The
independence condition is known to be false at the level of the joint
distributions of the test statistics, but that does not necessarily mean, for
the small significance levels involved in highly multiple hypothesis testing,
that the assumption leads to major errors. In this paper, we give conditions
under which the assumption of independence is valid. Specifically, we derive a
strong approximation that closely links the level exceedences of a dependent
``studentized process'' to those of a process of independent random variables.
Via this connection, it can be seen that in high-dimensional, low sample-size
cases, provided the sample size diverges faster than the logarithm of the
number of tests, the assumption of independent -tests is often justified.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ220 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Exact convergence rate and leading term in central limit theorem for student's t statistic
The leading term in the normal approximation to the distribution of Student's
t statistic is derived in a general setting, with the sole assumption being
that the sampled distribution is in the domain of attraction of a normal law.
The form of the leading term is shown to have its origin in the way in which
extreme data influence properties of the Studentized sum. The leading-term
approximation is used to give the exact rate of convergence in the central
limit theorem up to order n⁻¹/², where n denotes sample size. It is proved
that the exact rate uniformly on the whole real line is identical to the exact
rate on sets of just three points. Moreover, the exact rate is identical to
that for the non-Studentized sum when the latter is normalized for scale using
a truncated form of variance, but when the corresponding truncated centering
constant is omitted. Examples of characterizations of convergence rates are
also given. It is shown that, in some instances, their validity uniformly on
the whole real line is equivalent to their validity on just two symmetric
points
Effective temperature and glassy dynamics of active matter
A systematic expansion of the many-body master equation for active matter, in
which motors power configurational changes as in the cytoskeleton, is shown to
yield a description of the steady state and responses in terms of an effective
temperature. The effective temperature depends on the susceptibility of the
motors and a Peclet number which measures their strength relative to thermal
Brownian diffusion. The analytic prediction is shown to agree with previous
numerical simulations and experiments. The mapping also establishes a
description of aging in active matter that is also kinetically jammed.Comment: 2 figure
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