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The pressure function for infinite equilibrium measures
Assume that is a dynamical system and
is a potential such that the -invariant measure equivalent to
-conformal measure is infinite, but that there is an inducing scheme with a finite measure and polynomial tails
, . We give
conditions under which the pressure of for a perturbed potential
relates to the pressure of the induced system as , together with estimates for
the -error term. This extends results from Sarig to the setting of
infinite equilibrium states. We give several examples of such systems, thus
improving on the results of Lopes for the Pomeau-Manneville map with potential
, as well as on the results by Bruin & Todd on countably
piecewise linear unimodal Fibonacci maps. In addition, limit properties of the
family of measures as are studied and statistical
properties (correlation coefficients and arcsine laws) under the limit measure
are derived.Comment: Corrections in Section 8.2. Other minor modifications in the
presentatio
Reply to Murrell et al.: Noise matters
The concept of statistical “equitability” plays a central role in the 2011 paper by Reshef et al. (1). Formalizing equitability first requires formalizing the notion of a “noisy functional relationship,” that is, a relationship between two real variables, X and Y, having the form Y=f(X)+η, where f is a function and η is a noise term. Whether a dependence measure satisfies equitability strongly depends on what mathematical properties the noise term η is allowed to have: the narrower one’s definition of noise, the weaker the equitability criterion becomes
Sample size and positive false discovery rate control for multiple testing
The positive false discovery rate (pFDR) is a useful overall measure of
errors for multiple hypothesis testing, especially when the underlying goal is
to attain one or more discoveries. Control of pFDR critically depends on how
much evidence is available from data to distinguish between false and true
nulls. Oftentimes, as many aspects of the data distributions are unknown, one
may not be able to obtain strong enough evidence from the data for pFDR
control. This raises the question as to how much data are needed to attain a
target pFDR level. We study the asymptotics of the minimum number of
observations per null for the pFDR control associated with multiple Studentized
tests and tests, especially when the differences between false nulls and
true nulls are small. For Studentized tests, we consider tests on shifts or
other parameters associated with normal and general distributions. For
tests, we also take into account the effect of the number of covariates in
linear regression. The results show that in determining the minimum sample size
per null for pFDR control, higher order statistical properties of data are
important, and the number of covariates is important in tests to detect
regression effects.Comment: Published at http://dx.doi.org/10.1214/07-EJS045 in the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Confidence bands for a log-concave density
We present a new approach for inference about a log-concave distribution:
Instead of using the method of maximum likelihood, we propose to incorporate
the log-concavity constraint in an appropriate nonparametric confidence set for
the cdf . This approach has the advantage that it automatically provides a
measure of statistical uncertainty and it thus overcomes a marked limitation of
the maximum likelihood estimate. In particular, we show how to construct
confidence bands for the density that have a finite sample guaranteed
confidence level. The nonparametric confidence set for which we introduce
here has attractive computational and statistical properties: It allows to
bring modern tools from optimization to bear on this problem via difference of
convex programming, and it results in optimal statistical inference. We show
that the width of the resulting confidence bands converges at nearly the
parametric rate when the log density is -affine.Comment: Added more experiments, other minor change
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