494 research outputs found
Subsampling (weighted smooth) empirical copula processes
A key tool to carry out inference on the unknown copula when modeling a
continuous multivariate distribution is a nonparametric estimator known as the
empirical copula. One popular way of approximating its sampling distribution
consists of using the multiplier bootstrap. The latter is however characterized
by a high implementation cost. Given the rank-based nature of the empirical
copula, the classical empirical bootstrap of Efron does not appear to be a
natural alternative, as it relies on resamples which contain ties. The aim of
this work is to investigate the use of subsampling in the aforementioned
framework. The latter consists of basing the inference on statistic values
computed from subsamples of the initial data. One of its advantages in the
rank-based context under consideration is that the formed subsamples do not
contain ties. Another advantage is its asymptotic validity under minimalistic
conditions. In this work, we show the asymptotic validity of subsampling for
several (weighted, smooth) empirical copula processes both in the case of
serially independent observations and time series. In the former case,
subsampling is observed to be substantially better than the empirical bootstrap
and equivalent, overall, to the multiplier bootstrap in terms of finite-sample
performance.Comment: 34 pages, 5 figures, 4 + 8 table
A note on conditional versus joint unconditional weak convergence in bootstrap consistency results
The consistency of a bootstrap or resampling scheme is classically validated
by weak convergence of conditional laws. However, when working with stochastic
processes in the space of bounded functions and their weak convergence in the
Hoffmann-J{\o}rgensen sense, an obstacle occurs: due to possible
non-measurability, neither laws nor conditional laws are well-defined. Starting
from an equivalent formulation of weak convergence based on the bounded
Lipschitz metric, a classical circumvent is to formulate bootstrap consistency
in terms of the latter distance between what might be called a
\emph{conditional law} of the (non-measurable) bootstrap process and the law of
the limiting process. The main contribution of this note is to provide an
equivalent formulation of bootstrap consistency in the space of bounded
functions which is more intuitive and easy to work with. Essentially, the
equivalent formulation consists of (unconditional) weak convergence of the
original process jointly with two bootstrap replicates. As a by-product, we
provide two equivalent formulations of bootstrap consistency for statistics
taking values in separable metric spaces: the first in terms of (unconditional)
weak convergence of the statistic jointly with its bootstrap replicates, the
second in terms of convergence in probability of the empirical distribution
function of the bootstrap replicates. Finally, the asymptotic validity of
bootstrap-based confidence intervals and tests is briefly revisited, with
particular emphasis on the, in practice unavoidable, Monte Carlo approximation
of conditional quantiles.Comment: 21 pages, 1 Figur
Detecting distributional changes in samples of independent block maxima using probability weighted moments
The analysis of seasonal or annual block maxima is of interest in fields such
as hydrology, climatology or meteorology. In connection with the celebrated
method of block maxima, we study several tests that can be used to assess
whether the available series of maxima is identically distributed. It is
assumed that block maxima are independent but not necessarily generalized
extreme value distributed. The asymptotic null distributions of the test
statistics are investigated and the practical computation of approximate
p-values is addressed. Extensive Monte-Carlo simulations show the adequate
finite-sample behavior of the studied tests for a large number of realistic
data generating scenarios. Illustrations on several environmental datasets
conclude the work.Comment: 36 pages, 8 table
Distribution functions of linear combinations of lattice polynomials from the uniform distribution
We give the distribution functions, the expected values, and the moments of
linear combinations of lattice polynomials from the uniform distribution.
Linear combinations of lattice polynomials, which include weighted sums, linear
combinations of order statistics, and lattice polynomials, are actually those
continuous functions that reduce to linear functions on each simplex of the
standard triangulation of the unit cube. They are mainly used in aggregation
theory, combinatorial optimization, and game theory, where they are known as
discrete Choquet integrals and Lovasz extensions.Comment: 11 page
Goodness-of-fit testing based on a weighted bootstrap: A fast large-sample alternative to the parametric bootstrap
The process comparing the empirical cumulative distribution function of the
sample with a parametric estimate of the cumulative distribution function is
known as the empirical process with estimated parameters and has been
extensively employed in the literature for goodness-of-fit testing. The
simplest way to carry out such goodness-of-fit tests, especially in a
multivariate setting, is to use a parametric bootstrap. Although very easy to
implement, the parametric bootstrap can become very computationally expensive
as the sample size, the number of parameters, or the dimension of the data
increase. An alternative resampling technique based on a fast weighted
bootstrap is proposed in this paper, and is studied both theoretically and
empirically. The outcome of this work is a generic and computationally
efficient multiplier goodness-of-fit procedure that can be used as a
large-sample alternative to the parametric bootstrap. In order to approximately
determine how large the sample size needs to be for the parametric and weighted
bootstraps to have roughly equivalent powers, extensive Monte Carlo experiments
are carried out in dimension one, two and three, and for models containing up
to nine parameters. The computational gains resulting from the use of the
proposed multiplier goodness-of-fit procedure are illustrated on trivariate
financial data. A by-product of this work is a fast large-sample
goodness-of-fit procedure for the bivariate and trivariate t distribution whose
degrees of freedom are fixed.Comment: 26 pages, 5 tables, 1 figur
Large-sample tests of extreme-value dependence for multivariate copulas
Starting from the characterization of extreme-value copulas based on
max-stability, large-sample tests of extreme-value dependence for multivariate
copulas are studied. The two key ingredients of the proposed tests are the
empirical copula of the data and a multiplier technique for obtaining
approximate p-values for the derived statistics. The asymptotic validity of the
multiplier approach is established, and the finite-sample performance of a
large number of candidate test statistics is studied through extensive Monte
Carlo experiments for data sets of dimension two to five. In the bivariate
case, the rejection rates of the best versions of the tests are compared with
those of the test of Ghoudi, Khoudraji and Rivest (1998) recently revisited by
Ben Ghorbal, Genest and Neslehova (2009). The proposed procedures are
illustrated on bivariate financial data and trivariate geological data.Comment: 19 page
Semiparametric estimation of a two-component mixture of linear regressions in which one component is known
A new estimation method for the two-component mixture model introduced in
\cite{Van13} is proposed. This model consists of a two-component mixture of
linear regressions in which one component is entirely known while the
proportion, the slope, the intercept and the error distribution of the other
component are unknown. In spite of good performance for datasets of reasonable
size, the method proposed in \cite{Van13} suffers from a serious drawback when
the sample size becomes large as it is based on the optimization of a contrast
function whose pointwise computation requires O(n^2) operations. The range of
applicability of the method derived in this work is substantially larger as it
relies on a method-of-moments estimator free of tuning parameters whose
computation requires O(n) operations. From a theoretical perspective, the
asymptotic normality of both the estimator of the Euclidean parameter vector
and of the semiparametric estimator of the c.d.f.\ of the error is proved under
weak conditions not involving zero-symmetry assumptions. In addition, an
approximate confidence band for the c.d.f.\ of the error can be computed using
a weighted bootstrap whose asymptotic validity is proved. The finite-sample
performance of the resulting estimation procedure is studied under various
scenarios through Monte Carlo experiments. The proposed method is illustrated
on three real datasets of size , 51 and 176,343, respectively. Two
extensions of the considered model are discussed in the final section: a model
with an additional scale parameter for the first component, and a model with
more than one explanatory variable.Comment: 43 pages, 4 figures, 5 table
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