34 research outputs found
Robust Modeling Using Non-Elliptically Contoured Multivariate t Distributions
Models based on multivariate t distributions are widely applied to analyze
data with heavy tails. However, all the marginal distributions of the
multivariate t distributions are restricted to have the same degrees of
freedom, making these models unable to describe different marginal
heavy-tailedness. We generalize the traditional multivariate t distributions to
non-elliptically contoured multivariate t distributions, allowing for different
marginal degrees of freedom. We apply the non-elliptically contoured
multivariate t distributions to three widely-used models: the Heckman selection
model with different degrees of freedom for selection and outcome equations,
the multivariate Robit model with different degrees of freedom for marginal
responses, and the linear mixed-effects model with different degrees of freedom
for random effects and within-subject errors. Based on the Normal mixture
representation of our t distribution, we propose efficient Bayesian inferential
procedures for the model parameters based on data augmentation and parameter
expansion. We show via simulation studies and real examples that the
conclusions are sensitive to the existence of different marginal
heavy-tailedness
Optimal Linear Shrinkage Estimator for Large Dimensional Precision Matrix
In this work we construct an optimal shrinkage estimator for the precision
matrix in high dimensions. We consider the general asymptotics when the number
of variables and the sample size so
that . The precision matrix is estimated
directly, without inverting the corresponding estimator for the covariance
matrix. The recent results from the random matrix theory allow us to find the
asymptotic deterministic equivalents of the optimal shrinkage intensities and
estimate them consistently. The resulting distribution-free estimator has
almost surely the minimum Frobenius loss. Additionally, we prove that the
Frobenius norms of the inverse and of the pseudo-inverse sample covariance
matrices tend almost surely to deterministic quantities and estimate them
consistently. At the end, a simulation is provided where the suggested
estimator is compared with the estimators for the precision matrix proposed in
the literature. The optimal shrinkage estimator shows significant improvement
and robustness even for non-normally distributed data.Comment: 26 pages, 5 figures. This version includes the case c>1 with the
generalized inverse of the sample covariance matrix. The abstract was updated
accordingl
Stein Estimation for Spherically Symmetric Distributions: Recent Developments
This paper reviews advances in Stein-type shrinkage estimation for
spherically symmetric distributions. Some emphasis is placed on developing
intuition as to why shrinkage should work in location problems whether the
underlying population is normal or not. Considerable attention is devoted to
generalizing the "Stein lemma" which underlies much of the theoretical
development of improved minimax estimation for spherically symmetric
distributions. A main focus is on distributional robustness results in cases
where a residual vector is available to estimate an unknown scale parameter,
and, in particular, in finding estimators which are simultaneously generalized
Bayes and minimax over large classes of spherically symmetric distributions.
Some attention is also given to the problem of estimating a location vector
restricted to lie in a polyhedral cone.Comment: Published in at http://dx.doi.org/10.1214/10-STS323 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Reversing the Stein Effect
The Reverse Stein Effect is identified and illustrated: A statistician who
shrinks his/her data toward a point chosen without reliable knowledge about the
underlying value of the parameter to be estimated but based instead upon the
observed data will not be protected by the minimax property of shrinkage
estimators such as that of James and Stein, but instead will likely incur a
greater error than if shrinkage were not used.Comment: Published in at http://dx.doi.org/10.1214/09-STS278 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
An extended class of minimax generalized Bayes estimators of regression coefficients
We derive minimax generalized Bayes estimators of regression coefficients in
the general linear model with spherically symmetric errors under invariant
quadratic loss for the case of unknown scale. The class of estimators
generalizes the class considered in Maruyama and Strawderman (2005) to include
non-monotone shrinkage functions