27,234 research outputs found
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
On high-dimensional sign tests
Sign tests are among the most successful procedures in multivariate
nonparametric statistics. In this paper, we consider several testing problems
in multivariate analysis, directional statistics and multivariate time series
analysis, and we show that, under appropriate symmetry assumptions, the
fixed- multivariate sign tests remain valid in the high-dimensional case.
Remarkably, our asymptotic results are universal, in the sense that, unlike in
most previous works in high-dimensional statistics, may go to infinity in
an arbitrary way as does. We conduct simulations that (i) confirm our
asymptotic results, (ii) reveal that, even for relatively large , chi-square
critical values are to be favoured over the (asymptotically equivalent)
Gaussian ones and (iii) show that, for testing i.i.d.-ness against serial
dependence in the high-dimensional case, Portmanteau sign tests outperform
their competitors in terms of validity-robustness.Comment: Published at http://dx.doi.org/10.3150/15-BEJ710 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
High-dimensional tests for spherical location and spiked covariance
Rotationally symmetric distributions on the p-dimensional unit hypersphere,
extremely popular in directional statistics, involve a location parameter theta
that indicates the direction of the symmetry axis. The most classical way of
addressing the spherical location problem H_0:theta=theta_0, with theta_0 a
fixed location, is the so-called Watson test, which is based on the sample mean
of the observations. This test enjoys many desirable properties, but its
implementation requires the sample size n to be large compared to the dimension
p. This is a severe limitation, since more and more problems nowadays involve
high-dimensional directional data (e.g., in genetics or text mining). In this
work, we therefore introduce a modified Watson statistic that can cope with
high-dimensionality. We derive its asymptotic null distribution as both n and p
go to infinity. This is achieved in a universal asymptotic framework that
allows p to go to infinity arbitrarily fast (or slowly) as a function of n. We
further show that our results also provide high-dimensional tests for a problem
that has recently attracted much attention, namely that of testing that the
covariance matrix of a multinormal distribution has a "theta_0-spiked"
structure. Finally, a Monte Carlo simulation study corroborates our asymptotic
results
Modelling Directional Dispersion Through Hyperspherical Log- Splines
We introduce the directionally dispersed class of multivariate distributions, a generalisation of the elliptical class. By allowing dispersion of multivariate random variables to vary with direction it is possible to generate a very wide and flexible class of distributions. Directionally dispersed distributions are shown to have a simple form for their density, which extends a spherically symmetric density function by including a function D modelling directional dispersion. Under a mild condition, the class of distributions is shown to preserve both unimodality and moment existence. By adequately defining D, it is possible to generate skewed distributions. Using spline models on hyperspheres, we suggest a very general, yet practical, implementation for modelling directional dispersion in any dimension. Finally, we use the new class of distributions in a Bayesian regression setup and analyse the distributions of a set of biomedical measurements and a sample of U.S. manufacturing firms.Bayesian regression model, directional dispersion, elliptical distributions, existence of moments, modality, skewed distributions.
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