2,509 research outputs found
On the Coverage Bound Problem of Empirical Likelihood Methods For Time Series
The upper bounds on the coverage probabilities of the confidence regions
based on blockwise empirical likelihood [Kitamura (1997)] and nonstandard
expansive empirical likelihood [Nordman et al. (2013)] methods for time series
data are investigated via studying the probability for the violation of the
convex hull constraint. The large sample bounds are derived on the basis of the
pivotal limit of the blockwise empirical log-likelihood ratio obtained under
the fixed-b asymptotics, which has been recently shown to provide a more
accurate approximation to the finite sample distribution than the conventional
chi-square approximation. Our theoretical and numerical findings suggest that
both the finite sample and large sample upper bounds for coverage probabilities
are strictly less than one and the blockwise empirical likelihood confidence
region can exhibit serious undercoverage when (i) the dimension of moment
conditions is moderate or large; (ii) the time series dependence is positively
strong; or (iii) the block size is large relative to sample size. A similar
finite sample coverage problem occurs for the nonstandard expansive empirical
likelihood. To alleviate the coverage bound problem, we propose to penalize
both empirical likelihood methods by relaxing the convex hull constraint.
Numerical simulations and data illustration demonstrate the effectiveness of
our proposed remedies in terms of delivering confidence sets with more accurate
coverage
Regularized linear system identification using atomic, nuclear and kernel-based norms: the role of the stability constraint
Inspired by ideas taken from the machine learning literature, new
regularization techniques have been recently introduced in linear system
identification. In particular, all the adopted estimators solve a regularized
least squares problem, differing in the nature of the penalty term assigned to
the impulse response. Popular choices include atomic and nuclear norms (applied
to Hankel matrices) as well as norms induced by the so called stable spline
kernels. In this paper, a comparative study of estimators based on these
different types of regularizers is reported. Our findings reveal that stable
spline kernels outperform approaches based on atomic and nuclear norms since
they suitably embed information on impulse response stability and smoothness.
This point is illustrated using the Bayesian interpretation of regularization.
We also design a new class of regularizers defined by "integral" versions of
stable spline/TC kernels. Under quite realistic experimental conditions, the
new estimators outperform classical prediction error methods also when the
latter are equipped with an oracle for model order selection
Quasi-concave density estimation
Maximum likelihood estimation of a log-concave probability density is
formulated as a convex optimization problem and shown to have an equivalent
dual formulation as a constrained maximum Shannon entropy problem. Closely
related maximum Renyi entropy estimators that impose weaker concavity
restrictions on the fitted density are also considered, notably a minimum
Hellinger discrepancy estimator that constrains the reciprocal of the
square-root of the density to be concave. A limiting form of these estimators
constrains solutions to the class of quasi-concave densities.Comment: Published in at http://dx.doi.org/10.1214/10-AOS814 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Minimax and Adaptive Inference in Nonparametric Function Estimation
Since Stein's 1956 seminal paper, shrinkage has played a fundamental role in
both parametric and nonparametric inference. This article discusses minimaxity
and adaptive minimaxity in nonparametric function estimation. Three
interrelated problems, function estimation under global integrated squared
error, estimation under pointwise squared error, and nonparametric confidence
intervals, are considered. Shrinkage is pivotal in the development of both the
minimax theory and the adaptation theory. While the three problems are closely
connected and the minimax theories bear some similarities, the adaptation
theories are strikingly different. For example, in a sharp contrast to adaptive
point estimation, in many common settings there do not exist nonparametric
confidence intervals that adapt to the unknown smoothness of the underlying
function. A concise account of these theories is given. The connections as well
as differences among these problems are discussed and illustrated through
examples.Comment: Published in at http://dx.doi.org/10.1214/11-STS355 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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