1,069 research outputs found
Skew-symmetric distributions and Fisher information -- a tale of two densities
Skew-symmetric densities recently received much attention in the literature,
giving rise to increasingly general families of univariate and multivariate
skewed densities. Most of those families, however, suffer from the inferential
drawback of a potentially singular Fisher information in the vicinity of
symmetry. All existing results indicate that Gaussian densities (possibly after
restriction to some linear subspace) play a special and somewhat intriguing
role in that context. We dispel that widespread opinion by providing a full
characterization, in a general multivariate context, of the information
singularity phenomenon, highlighting its relation to a possible link between
symmetric kernels and skewing functions -- a link that can be interpreted as
the mismatch of two densities.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ346 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Flexible modelling in statistics: past, present and future
In times where more and more data become available and where the data exhibit
rather complex structures (significant departure from symmetry, heavy or light
tails), flexible modelling has become an essential task for statisticians as
well as researchers and practitioners from domains such as economics, finance
or environmental sciences. This is reflected by the wealth of existing
proposals for flexible distributions; well-known examples are Azzalini's
skew-normal, Tukey's -and-, mixture and two-piece distributions, to cite
but these. My aim in the present paper is to provide an introduction to this
research field, intended to be useful both for novices and professionals of the
domain. After a description of the research stream itself, I will narrate the
gripping history of flexible modelling, starring emblematic heroes from the
past such as Edgeworth and Pearson, then depict three of the most used flexible
families of distributions, and finally provide an outlook on future flexible
modelling research by posing challenging open questions.Comment: 27 pages, 4 figure
On the Independence Jeffreys prior for skew--symmetric models with applications
We study the Jeffreys prior of the skewness parameter of a general class of
scalar skew--symmetric models. It is shown that this prior is symmetric about
0, proper, and with tails under mild regularity conditions.
We also calculate the independence Jeffreys prior for the case with unknown
location and scale parameters. Sufficient conditions for the existence of the
corresponding posterior distribution are investigated for the case when the
sampling model belongs to the family of skew--symmetric scale mixtures of
normal distributions. The usefulness of these results is illustrated using the
skew--logistic model and two applications with real data
Bayesian Multivariate Regression Analysis with a New Class of Skewed Distributions
In this paper, we introduce a novel class of skewed multivariate distributions and, more generally, a method of building such a class on the basis of univariate skewed distributions. The method is based on a general linear transformation of a multidimensional random variable with independent components, each with a skewed distribution. Our proposed class of multivariate skewed distributions has a simple, intuitive form for the pdf, moment existence only depends on the existence of the moments of the underlying symmetric univariate distributions, and we avoid any conditioning on unobserved variables. In addition, we can freely allow for any mean and covariance structure in combination with any magnitude and direction of skewness. In order to deal with both skewness and fat tails, we introduce multivariate skewed regression models with fat tails, based on Student distributions. We present two main classes of such distributions, one of which is novel even under symmetry. Under standard non-informative priors on both regression and scale parameters, we derive conditions for propriety of the posterior and for existence of posterior moments. We describe MCMC samplers for conducting Bayesian inference and analyse two applications, one concerning the distribution of various measures of firm size and another on a set of biomedical data.Asymmetric distributions; Heavy tails; Linear regression model; Mardia's measure of skewness; Orthogonal matrices; Posterior propriety.
Statistical applications of the multivariate skew-normal distribution
Azzalini & Dalla Valle (1996) have recently discussed the multivariate
skew-normal distribution which extends the class of normal distributions by the
addition of a shape parameter. The first part of the present paper examines
further probabilistic properties of the distribution, with special emphasis on
aspects of statistical relevance. Inferential and other statistical issues are
discussed in the following part, with applications to some multivariate
statistics problems, illustrated by numerical examples. Finally, a further
extension is described which introduces a skewing factor of an elliptical
density.Comment: full-length version of the published paper, 32 pages, with 7 figures,
uses psfra
Skew-rotationally-symmetric distributions and related efficient inferential procedures
peer reviewedMost commonly used distributions on the unit hypersphere Sk−1={v∈Rk:v⊤v=1}, k≥2, assume that the data are rotationally symmetric about some direction θ∈Sk−1. However, there is empirical evidence that this assumption often fails to describe reality. We study in this paper a new class of skew-rotationally-symmetric distributions on Sk−1 that enjoy numerous good properties. We discuss the Fisher information structure of the model and derive efficient inferential procedures. In particular, we obtain the first semi-parametric test for rotational symmetry about a known direction. We also propose a second test for rotational symmetry, obtained through the definition of a new measure of skewness on the hypersphere. We investigate the finite-sample behavior of the new tests through a Monte Carlo simulation study. We conclude the paper with a discussion about some intriguing open questions related to our new models
Bayesian inference for skew-symmetric distributions
Skew-symmetric distributions are a popular family of flexible distributions that
conveniently model non-normal features such as skewness, kurtosis and multimodality.
Unfortunately, their frequentist inference poses several difficulties, which may be adequately
addressed by means of a Bayesian approach. This paper reviews the main prior distributions proposed
for the parameters of skew-symmetric distributions, with special emphasis on the skew-normal and
the skew-t distributions which are the most prominent skew-symmetric models. The paper focuses
on the univariate case in the absence of covariates, but more general models are also discussed
Distributions generated by perturbation of symmetry with emphasis on a multivariate skew distribution
A fairly general procedure is studied to perturbate a multivariate density
satisfying a weak form of multivariate symmetry, and to generate a whole set of
non-symmetric densities. The approach is general enough to encompass a number
of recent proposals in the literature, variously related to the skew normal
distribution. The special case of skew elliptical densities is examined in
detail, establishing connections with existing similar work. The final part of
the paper specializes further to a form of multivariate skew density.
Likelihood inference for this distribution is examined, and it is illustrated
with numerical examples.Comment: full-length version of the published paper, 31 pages with 9 figure
Skew-symmetric distributions and Fisher information
Hallin and Ley (2012) investigate and fully characterize the Fisher singularity phenomenon in univariate and multivariate families of skew-symmetric distributions. This paper proposes a refined analysis of the (univariate) Fisher degeneracy problem, showing that it can be more or less severe, inducing n1/4 (“simple singularity”), n1/6 (“double singularity”), or n1/8 (“triple singularity”) consistency rates for the skewness parameter.
We show, however, that simple singularity (yielding n1/4 consistency rates), if any singularity at all, is the rule, in the sense that double and triple singularities are possible for generalized skew-normal families only. We also show that higher-order
singularities, leading to worse-than-n1/8 rates, cannot occur
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