13,324 research outputs found
Towards Machine Wald
The past century has seen a steady increase in the need of estimating and
predicting complex systems and making (possibly critical) decisions with
limited information. Although computers have made possible the numerical
evaluation of sophisticated statistical models, these models are still designed
\emph{by humans} because there is currently no known recipe or algorithm for
dividing the design of a statistical model into a sequence of arithmetic
operations. Indeed enabling computers to \emph{think} as \emph{humans} have the
ability to do when faced with uncertainty is challenging in several major ways:
(1) Finding optimal statistical models remains to be formulated as a well posed
problem when information on the system of interest is incomplete and comes in
the form of a complex combination of sample data, partial knowledge of
constitutive relations and a limited description of the distribution of input
random variables. (2) The space of admissible scenarios along with the space of
relevant information, assumptions, and/or beliefs, tend to be infinite
dimensional, whereas calculus on a computer is necessarily discrete and finite.
With this purpose, this paper explores the foundations of a rigorous framework
for the scientific computation of optimal statistical estimators/models and
reviews their connections with Decision Theory, Machine Learning, Bayesian
Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty
Quantification and Information Based Complexity.Comment: 37 page
A One-Sample Test for Normality with Kernel Methods
We propose a new one-sample test for normality in a Reproducing Kernel
Hilbert Space (RKHS). Namely, we test the null-hypothesis of belonging to a
given family of Gaussian distributions. Hence our procedure may be applied
either to test data for normality or to test parameters (mean and covariance)
if data are assumed Gaussian. Our test is based on the same principle as the
MMD (Maximum Mean Discrepancy) which is usually used for two-sample tests such
as homogeneity or independence testing. Our method makes use of a special kind
of parametric bootstrap (typical of goodness-of-fit tests) which is
computationally more efficient than standard parametric bootstrap. Moreover, an
upper bound for the Type-II error highlights the dependence on influential
quantities. Experiments illustrate the practical improvement allowed by our
test in high-dimensional settings where common normality tests are known to
fail. We also consider an application to covariance rank selection through a
sequential procedure
Nonparametric inference on L\'evy measures and copulas
In this paper nonparametric methods to assess the multivariate L\'{e}vy
measure are introduced. Starting from high-frequency observations of a L\'{e}vy
process , we construct estimators for its tail integrals and the
Pareto-L\'{e}vy copula and prove weak convergence of these estimators in
certain function spaces. Given n observations of increments over intervals of
length , the rate of convergence is for
which is natural concerning inference on the L\'{e}vy measure. Besides
extensions to nonequidistant sampling schemes analytic properties of the
Pareto-L\'{e}vy copula which, to the best of our knowledge, have not been
mentioned before in the literature are provided as well. We conclude with a
short simulation study on the performance of our estimators and apply them to
real data.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1116 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Large-Scale Kernel Methods for Independence Testing
Representations of probability measures in reproducing kernel Hilbert spaces
provide a flexible framework for fully nonparametric hypothesis tests of
independence, which can capture any type of departure from independence,
including nonlinear associations and multivariate interactions. However, these
approaches come with an at least quadratic computational cost in the number of
observations, which can be prohibitive in many applications. Arguably, it is
exactly in such large-scale datasets that capturing any type of dependence is
of interest, so striking a favourable tradeoff between computational efficiency
and test performance for kernel independence tests would have a direct impact
on their applicability in practice. In this contribution, we provide an
extensive study of the use of large-scale kernel approximations in the context
of independence testing, contrasting block-based, Nystrom and random Fourier
feature approaches. Through a variety of synthetic data experiments, it is
demonstrated that our novel large scale methods give comparable performance
with existing methods whilst using significantly less computation time and
memory.Comment: 29 pages, 6 figure
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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