17,314 research outputs found
Multivariate Skew-t Distributions in Econometrics and Environmetrics
This dissertation is composed of three articles describing novel approaches for
analysis and modeling using multivariate skew-normal and skew-t distributions in
econometrics and environmetrics.
In the first article we introduce the Heckman selection-t model. Sample selection
arises often as a result of the partial observability of the outcome of interest in
a study. In the presence of sample selection, the observed data do not represent a
random sample from the population, even after controlling for explanatory variables.
Heckman introduced a sample-selection model to analyze such data and proposed a
full maximum likelihood estimation method under the assumption of normality. The
method was criticized in the literature because of its sensitivity to the normality assumption.
In practice, data, such as income or expenditure data, often violate the
normality assumption because of heavier tails. We first establish a new link between
sample-selection models and recently studied families of extended skew-elliptical distributions.
This then allows us to introduce a selection-t model, which models the
error distribution using a Student’s t distribution. We study its properties and investigate
the finite-sample performance of the maximum likelihood estimators for
this model. We compare the performance of the selection-t model to the Heckman
selection model and apply it to analyze ambulatory expenditures.
In the second article we introduce a family of multivariate log-skew-elliptical distributions,
extending the list of multivariate distributions with positive support. We
investigate their probabilistic properties such as stochastic representations, marginal
and conditional distributions, and existence of moments, as well as inferential properties.
We demonstrate, for example, that as for the log-t distribution, the positive
moments of the log-skew-t distribution do not exist. Our emphasis is on two special
cases, the log-skew-normal and log-skew-t distributions, which we use to analyze U.S.
precipitation data.
Many commonly used statistical methods assume that data are normally distributed.
This assumption is often violated in practice which prompted the development
of more flexible distributions. In the third article we describe two such multivariate
distributions, the skew-normal and the skew-t, and present commands for
fitting univariate and multivariate skew-normal and skew-t regressions in the statistical
software package Stata
Bayesian inference for the multivariate skew-normal model: a Population Monte Carlo approach
Frequentist and likelihood methods of inference based on the multivariate
skew-normal model encounter several technical difficulties with this model. In
spite of the popularity of this class of densities, there are no broadly
satisfactory solutions for estimation and testing problems. A general
population Monte Carlo algorithm is proposed which: 1) exploits the latent
structure stochastic representation of skew-normal random variables to provide
a full Bayesian analysis of the model and 2) accounts for the presence of
constraints in the parameter space. The proposed approach can be defined as
weakly informative, since the prior distribution approximates the actual
reference prior for the shape parameter vector. Results are compared with the
existing classical solutions and the practical implementation of the algorithm
is illustrated via a simulation study and a real data example. A generalization
to the matrix variate regression model with skew-normal error is also
presented
Regularized Multivariate Regression Models with Skew-\u3cem\u3et\u3c/em\u3e Error Distributions
We consider regularization of the parameters in multivariate linear regression models with the errors having a multivariate skew-t distribution. An iterative penalized likelihood procedure is proposed for constructing sparse estimators of both the regression coefficient and inverse scale matrices simultaneously. The sparsity is introduced through penalizing the negative log-likelihood by adding L1-penalties on the entries of the two matrices. Taking advantage of the hierarchical representation of skew-t distributions, and using the expectation conditional maximization (ECM) algorithm, we reduce the problem to penalized normal likelihood and develop a procedure to minimize the ensuing objective function. Using a simulation study the performance of the method is assessed, and the methodology is illustrated using a real data set with a 24-dimensional response vector
Non-Gaussian Geostatistical Modeling using (skew) t Processes
We propose a new model for regression and dependence analysis when addressing
spatial data with possibly heavy tails and an asymmetric marginal distribution.
We first propose a stationary process with marginals obtained through scale
mixing of a Gaussian process with an inverse square root process with Gamma
marginals. We then generalize this construction by considering a skew-Gaussian
process, thus obtaining a process with skew-t marginal distributions. For the
proposed (skew) process we study the second-order and geometrical
properties and in the case, we provide analytic expressions for the
bivariate distribution. In an extensive simulation study, we investigate the
use of the weighted pairwise likelihood as a method of estimation for the
process. Moreover we compare the performance of the optimal linear predictor of
the process versus the optimal Gaussian predictor. Finally, the
effectiveness of our methodology is illustrated by analyzing a georeferenced
dataset on maximum temperatures in Australi
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
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