1,700 research outputs found
Stein Estimation for Spherically Symmetric Distributions: Recent Developments
This paper reviews advances in Stein-type shrinkage estimation for
spherically symmetric distributions. Some emphasis is placed on developing
intuition as to why shrinkage should work in location problems whether the
underlying population is normal or not. Considerable attention is devoted to
generalizing the "Stein lemma" which underlies much of the theoretical
development of improved minimax estimation for spherically symmetric
distributions. A main focus is on distributional robustness results in cases
where a residual vector is available to estimate an unknown scale parameter,
and, in particular, in finding estimators which are simultaneously generalized
Bayes and minimax over large classes of spherically symmetric distributions.
Some attention is also given to the problem of estimating a location vector
restricted to lie in a polyhedral cone.Comment: Published in at http://dx.doi.org/10.1214/10-STS323 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Minimax Current Density Coil Design
'Coil design' is an inverse problem in which arrangements of wire are
designed to generate a prescribed magnetic field when energized with electric
current. The design of gradient and shim coils for magnetic resonance imaging
(MRI) are important examples of coil design. The magnetic fields that these
coils generate are usually required to be both strong and accurate. Other
electromagnetic properties of the coils, such as inductance, may be considered
in the design process, which becomes an optimization problem. The maximum
current density is additionally optimized in this work and the resultant coils
are investigated for performance and practicality. Coils with minimax current
density were found to exhibit maximally spread wires and may help disperse
localized regions of Joule heating. They also produce the highest possible
magnetic field strength per unit current for any given surface and wire size.
Three different flavours of boundary element method that employ different basis
functions (triangular elements with uniform current, cylindrical elements with
sinusoidal current and conic section elements with sinusoidal-uniform current)
were used with this approach to illustrate its generality.Comment: 24 pages, 6 figures, 2 tables. To appear in Journal of Physics D:
Applied Physic
Testing the isotropy of high energy cosmic rays using spherical needlets
For many decades, ultrahigh energy charged particles of unknown origin that
can be observed from the ground have been a puzzle for particle physicists and
astrophysicists. As an attempt to discriminate among several possible
production scenarios, astrophysicists try to test the statistical isotropy of
the directions of arrival of these cosmic rays. At the highest energies, they
are supposed to point toward their sources with good accuracy. However, the
observations are so rare that testing the distribution of such samples of
directional data on the sphere is nontrivial. In this paper, we choose a
nonparametric framework that makes weak hypotheses on the alternative
distributions and allows in turn to detect various and possibly unexpected
forms of anisotropy. We explore two particular procedures. Both are derived
from fitting the empirical distribution with wavelet expansions of densities.
We use the wavelet frame introduced by [SIAM J. Math. Anal. 38 (2006b) 574-594
(electronic)], the so-called needlets. The expansions are truncated at scale
indices no larger than some , and the distances between
those estimates and the null density are computed. One family of tests (called
Multiple) is based on the idea of testing the distance from the null for each
choice of , whereas the so-called PlugIn approach is
based on the single full expansion, but with thresholded wavelet
coefficients. We describe the practical implementation of these two procedures
and compare them to other methods in the literature. As alternatives to
isotropy, we consider both very simple toy models and more realistic
nonisotropic models based on Physics-inspired simulations. The Monte Carlo
study shows good performance of the Multiple test, even at moderate sample
size, for a wide sample of alternative hypotheses and for different choices of
the parameter . On the 69 most energetic events published by the
Pierre Auger Collaboration, the needlet-based procedures suggest statistical
evidence for anisotropy. Using several values for the parameters of the
methods, our procedures yield -values below 1%, but with uncontrolled
multiplicity issues. The flexibility of this method and the possibility to
modify it to take into account a large variety of extensions of the problem
make it an interesting option for future investigation of the origin of
ultrahigh energy cosmic rays.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS619 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Robust designs for series estimation
We discuss optimal design problems for a popular method of series estimation in regression problems. Commonly used design criteria are based on the generalized variance of the estimates of the coefficients in a truncated series expansion and do not take possible bias into account. We present a general perspective of constructing robust and e±cient designs for series estimators which is based on the integrated mean squared error criterion. A minimax approach is used to derive designs which are robust with respect to deviations caused by the bias and the possibility of heteroscedasticity. A special case results from the imposition of an unbiasedness constraint; the resulting unbiased designs are particularly simple, and easily implemented. Our results are illustrated by constructing robust designs for series estimation with spherical harmonic descriptors, Zernike polynomials and Chebyshev polynomials. --Chebyshev polynomials,direct estimation,minimax designs,robust designs,series estimation,spherical harmonic descriptors,unbiased design,Zernike polynomials
On the spherical convexity of quadratic functions
In this paper we study the spherical convexity of quadratic functions on
spherically convex sets. In particular, conditions characterizing the spherical
convexity of quadratic functions on spherical convex sets associated to the
positive orthants and Lorentz cones are given
Matched Filtering of Numerical Relativity Templates of Spinning Binary Black Holes
Tremendous progress has been made towards the solution of the
binary-black-hole problem in numerical relativity. The waveforms produced by
numerical relativity will play a role in gravitational wave detection as either
test-beds for analytic template banks or as template banks themselves. As the
parameter space explored by numerical relativity expands, the importance of
quantifying the effect that each parameter has on first the detection of
gravitational waves and then the parameter estimation of their sources
increases. In light of this, we present a study of equal-mass, spinning
binary-black-hole evolutions through matched filtering techniques commonly used
in data analysis. We study how the match between two numerical waveforms varies
with numerical resolution, initial angular momentum of the black holes and the
inclination angle between the source and the detector. This study is limited by
the fact that the spinning black-hole-binaries are oriented axially and the
waveforms only contain approximately two and a half orbits before merger. We
find that for detection purposes, spinning black holes require the inclusion of
the higher harmonics in addition to the dominant mode, a condition that becomes
more important as the black-hole-spins increase. In addition, we conduct a
preliminary investigation of how well a template of fixed spin and inclination
angle can detect target templates of arbitrary spin and inclination for the
axial case considered here
Fast Color Space Transformations Using Minimax Approximations
Color space transformations are frequently used in image processing,
graphics, and visualization applications. In many cases, these transformations
are complex nonlinear functions, which prohibits their use in time-critical
applications. In this paper, we present a new approach called Minimax
Approximations for Color-space Transformations (MACT).We demonstrate MACT on
three commonly used color space transformations. Extensive experiments on a
large and diverse image set and comparisons with well-known multidimensional
lookup table interpolation methods show that MACT achieves an excellent balance
among four criteria: ease of implementation, memory usage, accuracy, and
computational speed
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