2,455 research outputs found
A Path Algorithm for Constrained Estimation
Many least squares problems involve affine equality and inequality
constraints. Although there are variety of methods for solving such problems,
most statisticians find constrained estimation challenging. The current paper
proposes a new path following algorithm for quadratic programming based on
exact penalization. Similar penalties arise in regularization in model
selection. Classical penalty methods solve a sequence of unconstrained problems
that put greater and greater stress on meeting the constraints. In the limit as
the penalty constant tends to , one recovers the constrained solution.
In the exact penalty method, squared penalties are replaced by absolute value
penalties, and the solution is recovered for a finite value of the penalty
constant. The exact path following method starts at the unconstrained solution
and follows the solution path as the penalty constant increases. In the
process, the solution path hits, slides along, and exits from the various
constraints. Path following in lasso penalized regression, in contrast, starts
with a large value of the penalty constant and works its way downward. In both
settings, inspection of the entire solution path is revealing. Just as with the
lasso and generalized lasso, it is possible to plot the effective degrees of
freedom along the solution path. For a strictly convex quadratic program, the
exact penalty algorithm can be framed entirely in terms of the sweep operator
of regression analysis. A few well chosen examples illustrate the mechanics and
potential of path following.Comment: 26 pages, 5 figure
A Bayes method for a monotone hazard rate via S-paths
A class of random hazard rates, which is defined as a mixture of an indicator
kernel convolved with a completely random measure, is of interest. We provide
an explicit characterization of the posterior distribution of this mixture
hazard rate model via a finite mixture of S-paths. A closed and tractable Bayes
estimator for the hazard rate is derived to be a finite sum over S-paths. The
path characterization or the estimator is proved to be a Rao--Blackwellization
of an existing partition characterization or partition-sum estimator. This
accentuates the importance of S-paths in Bayesian modeling of monotone hazard
rates. An efficient Markov chain Monte Carlo (MCMC) method is proposed to
approximate this class of estimates. It is shown that S-path characterization
also exists in modeling with covariates by a proportional hazard model, and the
proposed algorithm again applies. Numerical results of the method are given to
demonstrate its practicality and effectiveness.Comment: Published at http://dx.doi.org/10.1214/009053606000000047 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On Degrees of Freedom of Projection Estimators with Applications to Multivariate Nonparametric Regression
In this paper, we consider the nonparametric regression problem with
multivariate predictors. We provide a characterization of the degrees of
freedom and divergence for estimators of the unknown regression function, which
are obtained as outputs of linearly constrained quadratic optimization
procedures, namely, minimizers of the least squares criterion with linear
constraints and/or quadratic penalties. As special cases of our results, we
derive explicit expressions for the degrees of freedom in many nonparametric
regression problems, e.g., bounded isotonic regression, multivariate
(penalized) convex regression, and additive total variation regularization. Our
theory also yields, as special cases, known results on the degrees of freedom
of many well-studied estimators in the statistics literature, such as ridge
regression, Lasso and generalized Lasso. Our results can be readily used to
choose the tuning parameter(s) involved in the estimation procedure by
minimizing the Stein's unbiased risk estimate. As a by-product of our analysis
we derive an interesting connection between bounded isotonic regression and
isotonic regression on a general partially ordered set, which is of independent
interest.Comment: 72 pages, 7 figures, Journal of the American Statistical Association
(Theory and Methods), 201
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