44 research outputs found
Weak Convergence of Probability Measures Revisited
The hypo-convergence of upper semicontinuous functions provides a natural framework for the study of the convergence of probability measures. This approach also yields some further characterizations of weak convergence and tightness
A Convergence of Bivariate Functions aimed at the Convergence of Saddle Functions
Epi/hypo-convergence is introduced from a variational viewpoint. The known topological properties are reviewed and extended. Finally, it is shown that the (partial) Legendre-Fenchel transform is bicontinuous with respect to the topology induced by epi/hypoconvergence on the space of convex-concave bivariate functions
Approximations of Semicontinuous Functions with Applications to Stochastic Optimization and Statistical Estimation
Upper semicontinuous (usc) functions arise in the analysis of maximization
problems, distributionally robust optimization, and function identification,
which includes many problems of nonparametric statistics. We establish that
every usc function is the limit of a hypo-converging sequence of piecewise
affine functions of the difference-of-max type and illustrate resulting
algorithmic possibilities in the context of approximate solution of
infinite-dimensional optimization problems. In an effort to quantify the ease
with which classes of usc functions can be approximated by finite collections,
we provide upper and lower bounds on covering numbers for bounded sets of usc
functions under the Attouch-Wets distance. The result is applied in the context
of stochastic optimization problems defined over spaces of usc functions. We
establish confidence regions for optimal solutions based on sample average
approximations and examine the accompanying rates of convergence. Examples from
nonparametric statistics illustrate the results
Variational Analysis of Constrained M-Estimators
We propose a unified framework for establishing existence of nonparametric
M-estimators, computing the corresponding estimates, and proving their strong
consistency when the class of functions is exceptionally rich. In particular,
the framework addresses situations where the class of functions is complex
involving information and assumptions about shape, pointwise bounds, location
of modes, height at modes, location of level-sets, values of moments, size of
subgradients, continuity, distance to a "prior" function, multivariate total
positivity, and any combination of the above. The class might be engineered to
perform well in a specific setting even in the presence of little data. The
framework views the class of functions as a subset of a particular metric space
of upper semicontinuous functions under the Attouch-Wets distance. In addition
to allowing a systematic treatment of numerous M-estimators, the framework
yields consistency of plug-in estimators of modes of densities, maximizers of
regression functions, level-sets of classifiers, and related quantities, and
also enables computation by means of approximating parametric classes. We
establish consistency through a one-sided law of large numbers, here extended
to sieves, that relaxes assumptions of uniform laws, while ensuring global
approximations even under model misspecification
When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs
In the past decades, weak convergence theory for stochastic processes has
become a standard tool for analyzing the asymptotic properties of various
statistics. Routinely, weak convergence is considered in the space of bounded
functions equipped with the supremum metric. However, there are cases when weak
convergence in those spaces fails to hold. Examples include empirical copula
and tail dependence processes and residual empirical processes in linear
regression models in case the underlying distributions lack a certain degree of
smoothness. To resolve the issue, a new metric for locally bounded functions is
introduced and the corresponding weak convergence theory is developed.
Convergence with respect to the new metric is related to epi- and
hypo-convergence and is weaker than uniform convergence. Still, for continuous
limits, it is equivalent to locally uniform convergence, whereas under mild
side conditions, it implies convergence. For the examples mentioned
above, weak convergence with respect to the new metric is established in
situations where it does not occur with respect to the supremum distance. The
results are applied to obtain asymptotic properties of resampling procedures
and goodness-of-fit tests.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1237 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A variational approach to a cumulative distribution function estimation problem under stochastic ambiguity
We propose a method for finding a cumulative distribution function (cdf) that
minimizes the (regularized) distance to a given cdf, while belonging to an
ambiguity set constructed relative to another cdf and, possibly, incorporating
soft information. Our method embeds the family of cdfs onto the space of upper
semicontinuous functions endowed with the hypo-distance. In this setting, we
present an approximation scheme based on epi-splines, defined as piecewise
polynomial functions, and use bounds for estimating the hypo-distance. Under
appropriate hypotheses, we guarantee that the cluster points corresponding to
the sequence of minimizers of the resulting approximating problems are
solutions to a limiting problem. In addition, we describe a large class of
functions that satisfy these hypotheses. The approximating method produces a
linear-programming-based approximation scheme, enabling us to develop an
algorithm from off-the-shelf solvers. The convergence of our proposed
approximation is illustrated by numerical examples for the bivariate case, one
of which entails a Lipschitz condition
Existence and uniqueness of viscosity solution for Hamilton-Jacobi equation with discontinuous coefficients dependent on time
The main result is a proof of the existence of a unique viscosity solution for Hamilton-Jacobi equation, where the hamiltonian is discontinuous with respect to variable, usually interpreted as the spatial one. Obtained generalized solution is continuous, but not necessarily differentiable