129 research outputs found
Statistical significance in high-dimensional linear models
We propose a method for constructing p-values for general hypotheses in a
high-dimensional linear model. The hypotheses can be local for testing a single
regression parameter or they may be more global involving several up to all
parameters. Furthermore, when considering many hypotheses, we show how to
adjust for multiple testing taking dependence among the p-values into account.
Our technique is based on Ridge estimation with an additional correction term
due to a substantial projection bias in high dimensions. We prove strong error
control for our p-values and provide sufficient conditions for detection: for
the former, we do not make any assumption on the size of the true underlying
regression coefficients while regarding the latter, our procedure might not be
optimal in terms of power. We demonstrate the method in simulated examples and
a real data application.Comment: Published in at http://dx.doi.org/10.3150/12-BEJSP11 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
On Bonferroni-Type Inequalities of the same Degree for the Probability of a Union
1 online resource (PDF, 16 pages
Partial spreads and vector space partitions
Constant-dimension codes with the maximum possible minimum distance have been
studied under the name of partial spreads in Finite Geometry for several
decades. Not surprisingly, for this subclass typically the sharpest bounds on
the maximal code size are known. The seminal works of Beutelspacher and Drake
\& Freeman on partial spreads date back to 1975, and 1979, respectively. From
then until recently, there was almost no progress besides some computer-based
constructions and classifications. It turns out that vector space partitions
provide the appropriate theoretical framework and can be used to improve the
long-standing bounds in quite a few cases. Here, we provide a historic account
on partial spreads and an interpretation of the classical results from a modern
perspective. To this end, we introduce all required methods from the theory of
vector space partitions and Finite Geometry in a tutorial style. We guide the
reader to the current frontiers of research in that field, including a detailed
description of the recent improvements.Comment: 30 pages, 1 tabl
Resource dependent branching processes and the envelope of societies
Since its early beginnings, mankind has put to test many different society
forms, and this fact raises a complex of interesting questions. The objective
of this paper is to present a general population model which takes essential
features of any society into account and which gives interesting answers on the
basis of only two natural hypotheses. One is that societies want to survive,
the second, that individuals in a society would, in general, like to increase
their standard of living. We start by presenting a mathematical model, which
may be seen as a particular type of a controlled branching process. All
conditions of the model are justified and interpreted. After several
preliminary results about societies in general we can show that two society
forms should attract particular attention, both from a qualitative and a
quantitative point of view. These are the so-called weakest-first society and
the strongest-first society. In particular we prove then that these two
societies stand out since they form an envelope of all possible societies in a
sense we will make precise. This result (the envelopment theorem) is seen as
significant because it is paralleled with precise survival criteria for the
enveloping societies. Moreover, given that one of the "limiting" societies can
be seen as an extreme form of communism, and the other one as being close to an
extreme version of capitalism, we conclude that, remarkably, humanity is close
to having already tested the limits.Comment: Published in at http://dx.doi.org/10.1214/13-AAP998 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Probabilistic Tools for the Analysis of Randomized Optimization Heuristics
This chapter collects several probabilistic tools that proved to be useful in
the analysis of randomized search heuristics. This includes classic material
like Markov, Chebyshev and Chernoff inequalities, but also lesser known topics
like stochastic domination and coupling or Chernoff bounds for geometrically
distributed random variables and for negatively correlated random variables.
Most of the results presented here have appeared previously, some, however,
only in recent conference publications. While the focus is on collecting tools
for the analysis of randomized search heuristics, many of these may be useful
as well in the analysis of classic randomized algorithms or discrete random
structures.Comment: 91 page
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