174 research outputs found
An overview of the goodness-of-fit test problem for copulas
We review the main "omnibus procedures" for goodness-of-fit testing for
copulas: tests based on the empirical copula process, on probability integral
transformations, on Kendall's dependence function, etc, and some corresponding
reductions of dimension techniques. The problems of finding asymptotic
distribution-free test statistics and the calculation of reliable p-values are
discussed. Some particular cases, like convenient tests for time-dependent
copulas, for Archimedean or extreme-value copulas, etc, are dealt with.
Finally, the practical performances of the proposed approaches are briefly
summarized
Goodness-of-fit Tests For Elliptical And Independent Copulas Through Projection Pursuit
Two goodness-of-fit tests for copulas are being investigated. The first one
deals with the case of elliptical copulas and the second one deals with
independent copulas. These tests result from the expansion of the projection
pursuit methodology we will introduce in the present article. This method
enables us to determine on which axis system these copulas lie as well as the
exact value of these very copulas in the basis formed by the axes previously
determined irrespective of their value in their canonical basis. Simulations
are also presented as well as an application to real datasets.Comment: 31 page
Parametric versus nonparametric: the fitness coefficient
The fitness coefficient, introduced in this paper, results from a competition
between parametric and nonparametric density estimators within the likelihood
of the data. As illustrated on several real datasets, the fitness coefficient
generally agrees with p-values but is easier to compute and interpret. Namely,
the fitness coefficient can be interpreted as the proportion of data coming
from the parametric model. Moreover, the fitness coefficient can be used to
build a semiparamteric compromise which improves inference over the parametric
and nonparametric approaches. From a theoretical perspective, the fitness
coefficient is shown to converge in probability to one if the model is true and
to zero if the model is false. From a practical perspective, the utility of the
fitness coefficient is illustrated on real and simulated datasets
Extreme value copula estimation based on block maxima of a multivariate stationary time series
The core of the classical block maxima method consists of fitting an extreme
value distribution to a sample of maxima over blocks extracted from an
underlying series. In asymptotic theory, it is usually postulated that the
block maxima are an independent random sample of an extreme value distribution.
In practice however, block sizes are finite, so that the extreme value
postulate will only hold approximately. A more accurate asymptotic framework is
that of a triangular array of block maxima, the block size depending on the
size of the underlying sample in such a way that both the block size and the
number of blocks within that sample tend to infinity. The copula of the vector
of componentwise maxima in a block is assumed to converge to a limit, which,
under mild conditions, is then necessarily an extreme value copula. Under this
setting and for absolutely regular stationary sequences, the empirical copula
of the sample of vectors of block maxima is shown to be a consistent and
asymptotically normal estimator for the limiting extreme value copula.
Moreover, the empirical copula serves as a basis for rank-based, nonparametric
estimation of the Pickands dependence function of the extreme value copula. The
results are illustrated by theoretical examples and a Monte Carlo simulation
study.Comment: 34 page
Nonparametric independence testing via mutual information
We propose a test of independence of two multivariate random vectors, given a
sample from the underlying population. Our approach, which we call MINT, is
based on the estimation of mutual information, whose decomposition into joint
and marginal entropies facilitates the use of recently-developed efficient
entropy estimators derived from nearest neighbour distances. The proposed
critical values, which may be obtained from simulation (in the case where one
marginal is known) or resampling, guarantee that the test has nominal size, and
we provide local power analyses, uniformly over classes of densities whose
mutual information satisfies a lower bound. Our ideas may be extended to
provide a new goodness-of-fit tests of normal linear models based on assessing
the independence of our vector of covariates and an appropriately-defined
notion of an error vector. The theory is supported by numerical studies on both
simulated and real data.EPSRC
Leverhulme Trust
SIMS fun
Statistical Aspects of Wasserstein Distances
Wasserstein distances are metrics on probability distributions inspired by
the problem of optimal mass transportation. Roughly speaking, they measure the
minimal effort required to reconfigure the probability mass of one distribution
in order to recover the other distribution. They are ubiquitous in mathematics,
with a long history that has seen them catalyse core developments in analysis,
optimization, and probability. Beyond their intrinsic mathematical richness,
they possess attractive features that make them a versatile tool for the
statistician: they can be used to derive weak convergence and convergence of
moments, and can be easily bounded; they are well-adapted to quantify a natural
notion of perturbation of a probability distribution; and they seamlessly
incorporate the geometry of the domain of the distributions in question, thus
being useful for contrasting complex objects. Consequently, they frequently
appear in the development of statistical theory and inferential methodology,
and have recently become an object of inference in themselves. In this review,
we provide a snapshot of the main concepts involved in Wasserstein distances
and optimal transportation, and a succinct overview of some of their many
statistical aspects.Comment: Official version available at
https://www.annualreviews.org/doi/full/10.1146/annurev-statistics-030718-10493
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