38,375 research outputs found
EMMIXcskew: an R Package for the Fitting of a Mixture of Canonical Fundamental Skew t-Distributions
This paper presents an R package EMMIXcskew for the fitting of the canonical
fundamental skew t-distribution (CFUST) and finite mixtures of this
distribution (FM-CFUST) via maximum likelihood (ML). The CFUST distribution
provides a flexible family of models to handle non-normal data, with parameters
for capturing skewness and heavy-tails in the data. It formally encompasses the
normal, t, and skew-normal distributions as special and/or limiting cases. A
few other versions of the skew t-distributions are also nested within the CFUST
distribution. In this paper, an Expectation-Maximization (EM) algorithm is
described for computing the ML estimates of the parameters of the FM-CFUST
model, and different strategies for initializing the algorithm are discussed
and illustrated. The methodology is implemented in the EMMIXcskew package, and
examples are presented using two real datasets. The EMMIXcskew package contains
functions to fit the FM-CFUST model, including procedures for generating
different initial values. Additional features include random sample generation
and contour visualization in 2D and 3D
Exchangeable random measures
Let A be a standard Borel space, and consider the space A^{\bbN^{(k)}} of
A-valued arrays indexed by all size-k subsets of \bbN. This paper concerns
random measures on such a space whose laws are invariant under the natural
action of permutations of \bbN. The main result is a representation theorem for
such `exchangeable' random measures, obtained using the classical
representation theorems for exchangeable arrays due to de Finetti, Hoover,
Aldous and Kallenberg.
After proving this representation, two applications of exchangeable random
measures are given. The first is a short new proof of the Dovbysh-Sudakov
Representation Theorem for exchangeable PSD matrices. The second is in the
formulation of a natural class of limit objects for dilute mean-field spin
glass models, retaining more information than just the limiting Gram-de Finetti
matrix used in the study of the Sherrington-Kirkpatrick model.Comment: 24 pages. [4/23/2013:] Re-written for clarity, but no conceptual
changes. [9/12/2013:] Slightly re-written to incorporate referee suggestions.
[7/8/15:] Published version available at
http://projecteuclid.org/euclid.aihp/143575923
Four moments theorems on Markov chains
We obtain quantitative Four Moments Theorems establishing convergence
of the laws of elements of a Markov chaos to a Pearson distribution,
where the only assumptionwemake on the Pearson distribution is that it admits
four moments. While in general one cannot use moments to establish convergence
to a heavy-tailed distributions, we provide a context in which only the
first four moments suffices. These results are obtained by proving a general
carré du champ bound on the distance between laws of random variables in the
domain of a Markov diffusion generator and invariant measures of diffusions.
For elements of a Markov chaos, this bound can be reduced to just the first four
moments.First author draf
Four moments theorems on Markov chaos
We obtain quantitative Four Moments Theorems establishing convergence of the
laws of elements of a Markov chaos to a Pearson distribution, where the only
assumption we make on the Pearson distribution is that it admits four moments.
While in general one cannot use moments to establish convergence to a
heavy-tailed distributions, we provide a context in which only the first four
moments suffices. These results are obtained by proving a general carr\'e du
champ bound on the distance between laws of random variables in the domain of a
Markov diffusion generator and invariant measures of diffusions. For elements
of a Markov chaos, this bound can be reduced to just the first four moments.Comment: 24 page
Conjugate Bayes for probit regression via unified skew-normal distributions
Regression models for dichotomous data are ubiquitous in statistics. Besides
being useful for inference on binary responses, these methods serve also as
building blocks in more complex formulations, such as density regression,
nonparametric classification and graphical models. Within the Bayesian
framework, inference proceeds by updating the priors for the coefficients,
typically set to be Gaussians, with the likelihood induced by probit or logit
regressions for the responses. In this updating, the apparent absence of a
tractable posterior has motivated a variety of computational methods, including
Markov Chain Monte Carlo routines and algorithms which approximate the
posterior. Despite being routinely implemented, Markov Chain Monte Carlo
strategies face mixing or time-inefficiency issues in large p and small n
studies, whereas approximate routines fail to capture the skewness typically
observed in the posterior. This article proves that the posterior distribution
for the probit coefficients has a unified skew-normal kernel, under Gaussian
priors. Such a novel result allows efficient Bayesian inference for a wide
class of applications, especially in large p and small-to-moderate n studies
where state-of-the-art computational methods face notable issues. These
advances are outlined in a genetic study, and further motivate the development
of a wider class of conjugate priors for probit models along with methods to
obtain independent and identically distributed samples from the unified
skew-normal posterior
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