1,703 research outputs found
Aggregation of Affine Estimators
We consider the problem of aggregating a general collection of affine
estimators for fixed design regression. Relevant examples include some commonly
used statistical estimators such as least squares, ridge and robust least
squares estimators. Dalalyan and Salmon (2012) have established that, for this
problem, exponentially weighted (EW) model selection aggregation leads to sharp
oracle inequalities in expectation, but similar bounds in deviation were not
previously known. While results indicate that the same aggregation scheme may
not satisfy sharp oracle inequalities with high probability, we prove that a
weaker notion of oracle inequality for EW that holds with high probability.
Moreover, using a generalization of the newly introduced -aggregation scheme
we also prove sharp oracle inequalities that hold with high probability.
Finally, we apply our results to universal aggregation and show that our
proposed estimator leads simultaneously to all the best known bounds for
aggregation, including -aggregation, , with high
probability
Sharp Oracle Inequalities for Aggregation of Affine Estimators
We consider the problem of combining a (possibly uncountably infinite) set of
affine estimators in non-parametric regression model with heteroscedastic
Gaussian noise. Focusing on the exponentially weighted aggregate, we prove a
PAC-Bayesian type inequality that leads to sharp oracle inequalities in
discrete but also in continuous settings. The framework is general enough to
cover the combinations of various procedures such as least square regression,
kernel ridge regression, shrinking estimators and many other estimators used in
the literature on statistical inverse problems. As a consequence, we show that
the proposed aggregate provides an adaptive estimator in the exact minimax
sense without neither discretizing the range of tuning parameters nor splitting
the set of observations. We also illustrate numerically the good performance
achieved by the exponentially weighted aggregate
Optimal exponential bounds for aggregation of estimators for the Kullback-Leibler loss
We study the problem of model selection type aggregation with respect to the
Kullback-Leibler divergence for various probabilistic models. Rather than
considering a convex combination of the initial estimators ,
our aggregation procedures rely on the convex combination of the logarithms of
these functions. The first method is designed for probability density
estimation as it gives an aggregate estimator that is also a proper density
function, whereas the second method concerns spectral density estimation and
has no such mass-conserving feature. We select the aggregation weights based on
a penalized maximum likelihood criterion. We give sharp oracle inequalities
that hold with high probability, with a remainder term that is decomposed into
a bias and a variance part. We also show the optimality of the remainder terms
by providing the corresponding lower bound results.Comment: 25 page
Confidence Intervals for Maximin Effects in Inhomogeneous Large-Scale Data
One challenge of large-scale data analysis is that the assumption of an
identical distribution for all samples is often not realistic. An optimal
linear regression might, for example, be markedly different for distinct groups
of the data. Maximin effects have been proposed as a computationally attractive
way to estimate effects that are common across all data without fitting a
mixture distribution explicitly. So far just point estimators of the common
maximin effects have been proposed in Meinshausen and B\"uhlmann (2014). Here
we propose asymptotically valid confidence regions for these effects
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