44 research outputs found
The Majorant Lyapunov Equation: A Nonnegative Matrix Equation for Robust Stability and Performance of Large Scale Systems
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/57872/1/MajorantTAC1987.pd
Extremal functions in de Branges and Euclidean spaces
In this work we obtain optimal majorants and minorants of exponential type
for a wide class of radial functions on . These extremal
functions minimize the -distance to
the original function, where is a free parameter. To achieve this
result we develop new interpolation tools to solve an associated extremal
problem for the exponential function , where , in the general framework of de Branges
spaces of entire functions. We then specialize the construction to a particular
family of homogeneous de Branges spaces to approach the multidimensional
Euclidean case. Finally, we extend the result from the exponential function to
a class of subordinated radial functions via integration on the parameter
against suitable measures. Applications of the results presented
here include multidimensional versions of Hilbert-type inequalities, extremal
one-sided approximations by trigonometric polynomials for a class of even
periodic functions and extremal one-sided approximations by polynomials for a
class of functions on the sphere with an axis of symmetry
Multiplier bootstrap of tail copulas with applications
For the problem of estimating lower tail and upper tail copulas, we propose
two bootstrap procedures for approximating the distribution of the
corresponding empirical tail copulas. The first method uses a multiplier
bootstrap of the empirical tail copula process and requires estimation of the
partial derivatives of the tail copula. The second method avoids this
estimation problem and uses multipliers in the two-dimensional empirical
distribution function and in the estimates of the marginal distributions. For
both multiplier bootstrap procedures, we prove consistency. For these
investigations, we demonstrate that the common assumption of the existence of
continuous partial derivatives in the the literature on tail copula estimation
is so restrictive, such that the tail copula corresponding to tail independence
is the only tail copula with this property. Moreover, we are able to solve this
problem and prove weak convergence of the empirical tail copula process under
nonrestrictive smoothness assumptions that are satisfied for many commonly used
models. These results are applied in several statistical problems, including
minimum distance estimation and goodness-of-fit testing.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ425 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Recommended from our members
Poincaré and the Three Body Problem
The purpose of the thesis is to present an account of Henri Poincare's famous memoir on the three body problem, the final version of which was published in Acta Mathematica in 1890 as the prize-winning entry in King Oscar II's 60th birthday competition. The memoir is reknowned both for its role in providing the foundations for Poincare's celebrated three volume Méthodes Nouvelles de la Mécanique Céleste, and for containing the first mathematical description of chaotic behaviour in a dynamical system.
A historical context is provided both through consideration of the problem itself and through a discussion of Poincaré's earlier work which relates to the mathematics developed in the memoir. The organisation of the Oscar competition, which was undertaken by Gösta Mittag-Leffler, is also described. This not only provides an insight into the late 19th century European mathematical community but also reveals that after the prize had been awarded Poincare found an important error in his work and substantially revised the memoir prior to its publication in Acta. The discovery of a printed version of the original memoir personally annotated by Poincaré has allowed for a detailed comparative study of the mathematics contained in both versions of the memoir. The error is explained and it is shown that it was only as a result of its correction that Poincaré discovered the chaotic behaviour now associated with the memoir.
The contemporary reception of the memoir is discussed and Poincaré's subsequent work in celestial mechanics and related topics is examined. Through the consideration of sources up to 1920 the influence and impact of the memoir on the progress of the three body problem and on dynamics in general is assessed
Proof mining in metric fixed point theory and ergodic theory
In this survey we present some recent applications of proof mining to the
fixed point theory of (asymptotically) nonexpansive mappings and to the
metastability (in the sense of Terence Tao) of ergodic averages in uniformly
convex Banach spaces.Comment: appeared as OWP 2009-05, Oberwolfach Preprints; 71 page
A test for Archimedeanity in bivariate copula models
We propose a new test for the hypothesis that a bivariate copula is an
Archimedean copula. The test statistic is based on a combination of two
measures resulting from the characterization of Archimedean copulas by the
property of associativity and by a strict upper bound on the diagonal by the
Fr\'echet-upper bound. We prove weak convergence of this statistic and show
that the critical values of the corresponding test can be determined by the
multiplier bootstrap method. The test is shown to be consistent against all
departures from Archimedeanity if the copula satisfies weak smoothness
assumptions. A simulation study is presented which illustrates the finite
sample properties of the new test.Comment: 18 pages, 2 figure