591 research outputs found
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
High angular resolution diffusion imaging data is the observed characteristic
function for the local diffusion of water molecules in tissue. This data is
used to infer structural information in brain imaging. Nonparametric scalar
measures are proposed to summarize such data, and to locally characterize
spatial features of the diffusion probability density function (PDF), relying
on the geometry of the characteristic function. Summary statistics are defined
so that their distributions are, to first-order, both independent of nuisance
parameters and also analytically tractable. The dominant direction of the
diffusion at a spatial location (voxel) is determined, and a new set of axes
are introduced in Fourier space. Variation quantified in these axes determines
the local spatial properties of the diffusion density. Nonparametric hypothesis
tests for determining whether the diffusion is unimodal, isotropic or
multi-modal are proposed. More subtle characteristics of white-matter
microstructure, such as the degree of anisotropy of the PDF and symmetry
compared with a variety of asymmetric PDF alternatives, may be ascertained
directly in the Fourier domain without parametric assumptions on the form of
the diffusion PDF. We simulate a set of diffusion processes and characterize
their local properties using the newly introduced summaries. We show how
complex white-matter structures across multiple voxels exhibit clear
ellipsoidal and asymmetric structure in simulation, and assess the performance
of the statistics in clinically-acquired magnetic resonance imaging data.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS441 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Test for Multimodality of Regression Derivatives with an Application to Nonparametric Growth Regressions
This paper presents a method to test for multimodality of an estimated kernel density of parameter estimates from a local-linear least-squares regression derivative. The procedure is laid out in seven simple steps and a suggestion for implementation is proposed. A Monte Carlo exercise is used to examine the finite sample properties of the test along with those from a calibrated version of it which corrects for the conservative nature of Silverman-type tests. The test is included in a study on nonparametric growth regressions. The results show that in the estimation of unconditional β-convergence, the distribution of the parameter estimates is multimodal with one mode in the negative region (primarily OECD economies) and possibly two modes in the positive region (primarily non-OECD economies) of the parameter estimates. The results for conditional β-convergence show that the density is predominantly negative and unimodal. Finally, the application attempts to determine why particular observations posess positive marginal effects on initial income in both the unconditional and conditional frameworks.Nonparametric Kernel; Convergence; Modality Tests
A Distribution Dynamics Approach to Regional GDP Convergence in Unified Germany
This paper uses nonparametric techniques to study GDP convergence across German labor market regions and counties during the period 1992-2004. The main result is that regional convergence in unified Germany has been substantial. In the first years after German unification the distribution of GDP has been characterized by a pronounced bimodality. The dispersion of the GDP distribution has become substantially smaller over time. Although some bimodality remains in most recent years, this bimodality is weak in comparison to previous years. Nevertheless, disparities among regions located in the Eastern and Western part of the country are still apparent.regional convergence, distribution dynamics, nonparametric econometrics, stochastic kernel
Bump hunting with non-Gaussian kernels
It is well known that the number of modes of a kernel density estimator is
monotone nonincreasing in the bandwidth if the kernel is a Gaussian density.
There is numerical evidence of nonmonotonicity in the case of some non-Gaussian
kernels, but little additional information is available. The present paper
provides theoretical and numerical descriptions of the extent to which the
number of modes is a nonmonotone function of bandwidth in the case of general
compactly supported densities. Our results address popular kernels used in
practice, for example, the Epanechnikov, biweight and triweight kernels, and
show that in such cases nonmonotonicity is present with strictly positive
probability for all sample sizes n\geq3. In the Epanechnikov and biweight cases
the probability of nonmonotonicity equals 1 for all n\geq2. Nevertheless, in
spite of the prevalence of lack of monotonicity revealed by these results, it
is shown that the notion of a critical bandwidth (the smallest bandwidth above
which the number of modes is guaranteed to be monotone) is still well defined.
Moreover, just as in the Gaussian case, the critical bandwidth is of the same
size as the bandwidth that minimises mean squared error of the density
estimator. These theoretical results, and new numerical evidence, show that the
main effects of nonmonotonicity occur for relatively small bandwidths, and have
negligible impact on many aspects of bump hunting.Comment: Published at http://dx.doi.org/10.1214/009053604000000715 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Nonlinear dynamics in welfare and the evolution of world inequality
The paper proposes a measure of countries' welfare based on individuals' lifetime utility and applies it to a large sample of countries in the period 1960-2000. Even though welfare inequality across countries appeared stable, the distribution dynamics points out the emergence of three clusters. Such tendencies to polarization shall strengthen in the future. In terms of the world population distribution, welfare inequality decreased as the result of the decline in inequality of both per capita GDP and life expectancy, but this downward trend should be reverted hereafter. Finally, a polarization pattern emerged, which is expected to further intensify in the future.distribution of welfare, nonparametric estimation, polarization, distribution dynamics, inequality
Robust nonparametric inference for the median
We consider the problem of constructing robust nonparametric confidence
intervals and tests of hypothesis for the median when the data distribution is
unknown and the data may contain a small fraction of contamination. We propose
a modification of the sign test (and its associated confidence interval) which
attains the nominal significance level (probability coverage) for any
distribution in the contamination neighborhood of a continuous distribution. We
also define some measures of robustness and efficiency under contamination for
confidence intervals and tests. These measures are computed for the proposed
procedures.Comment: Published at http://dx.doi.org/10.1214/009053604000000634 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A class of optimal tests for symmetry based on local Edgeworth approximations
The objective of this paper is to provide, for the problem of univariate
symmetry (with respect to specified or unspecified location), a concept of
optimality, and to construct tests achieving such optimality. This requires
embedding symmetry into adequate families of asymmetric (local) alternatives.
We construct such families by considering non-Gaussian generalizations of
classical first-order Edgeworth expansions indexed by a measure of skewness
such that (i) location, scale and skewness play well-separated roles
(diagonality of the corresponding information matrices) and (ii) the classical
tests based on the Pearson--Fisher coefficient of skewness are optimal in the
vicinity of Gaussian densities.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ298 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Statistical inference for inverse problems
In this paper we study statistical inference for certain inverse problems. We go beyond mere estimation purposes and review and develop the construction of confidence intervals and confidence bands in some inverse problems, including deconvolution and the backward heat equation. Further, we discuss the construction of certain hypothesis tests, in particular concerning the number of local maxima of the unknown function. The methods are illustrated in a case study, where we analyze the distribution of heliocentric escape velocities of galaxies in the Centaurus galaxy cluster, and provide statistical evidence for its bimodality. --Asymptotic normality,confidence interval,deconvolution,heat equation,modality,statistical inference,statistical inverse problem
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