68 research outputs found
Strong Approximation of Empirical Copula Processes by Gaussian Processes
We provide the strong approximation of empirical copula processes by a
Gaussian process. In addition we establish a strong approximation of the
smoothed empirical copula processes and a law of iterated logarithm
A test for Archimedeanity in bivariate copula models
We propose a new test for the hypothesis that a bivariate copula is an
Archimedean copula. The test statistic is based on a combination of two
measures resulting from the characterization of Archimedean copulas by the
property of associativity and by a strict upper bound on the diagonal by the
Fr\'echet-upper bound. We prove weak convergence of this statistic and show
that the critical values of the corresponding test can be determined by the
multiplier bootstrap method. The test is shown to be consistent against all
departures from Archimedeanity if the copula satisfies weak smoothness
assumptions. A simulation study is presented which illustrates the finite
sample properties of the new test.Comment: 18 pages, 2 figure
Resampling Procedures with Empirical Beta Copulas
The empirical beta copula is a simple but effective smoother of the empirical
copula. Because it is a genuine copula, from which, moreover, it is
particularly easy to sample, it is reasonable to expect that resampling
procedures based on the empirical beta copula are expedient and accurate. In
this paper, after reviewing the literature on some bootstrap approximations for
the empirical copula process, we first show the asymptotic equivalence of
several bootstrapped processes related to the empirical copula and empirical
beta copula. Then we investigate the finite-sample properties of resampling
schemes based on the empirical (beta) copula by Monte Carlo simulation. More
specifically, we consider interval estimation for some functionals such as rank
correlation coefficients and dependence parameters of several well-known
families of copulas, constructing confidence intervals by several methods and
comparing their accuracy and efficiency. We also compute the actual size and
power of symmetry tests based on several resampling schemes for the empirical
copula and empirical beta copula.Comment: 22 pages, 8 table
Asymptotics of empirical copula processes under non-restrictive smoothness assumptions
Weak convergence of the empirical copula process is shown to hold under the
assumption that the first-order partial derivatives of the copula exist and are
continuous on certain subsets of the unit hypercube. The assumption is
non-restrictive in the sense that it is needed anyway to ensure that the
candidate limiting process exists and has continuous trajectories. In addition,
resampling methods based on the multiplier central limit theorem, which require
consistent estimation of the first-order derivatives, continue to be valid.
Under certain growth conditions on the second-order partial derivatives that
allow for explosive behavior near the boundaries, the almost sure rate in
Stute's representation of the empirical copula process can be recovered. The
conditions are verified, for instance, in the case of the Gaussian copula with
full-rank correlation matrix, many Archimedean copulas, and many extreme-value
copulas.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ387 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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 multivariate density estimation for positive data using copulas
In this paper we estimate density functions for positive multivariate data. We propose a semiparametric approach. The estimator combines gamma kernels or local linear kernels, also called boundary kernels, for the estimation of the marginal densities with semiparametric copulas to model the dependence. This semiparametric approach is robust both to the well known boundary bias problem and the curse of dimensionality problem. We derive the mean integrated squared error properties, including the rate of convergence, the uniform strong consistency and the asymptotic normality. A simulation study investigates the finite sample performance of the estimator. We find that univariate least squares cross validation, to choose the bandwidth for the estimation of the marginal densities, works well and that the estimator we propose performs very well also for data with unbounded support. Applications in the field of finance are provided.asymptotic properties, asymmetric kernels, boundary bias, copula, curse of dimension, least squares cross validation
Semiparametric Multivariate Density Estimation for Positive Data Using Copulas.
In this paper we estimate density functions for positive multivariate data. We propose a semiparametric approach. The estimator combines gamma kernels or local linear kernels, also called boundary kernels, for the estimation of the marginal densities with semiparametric copulas to model the dependence. This semiparametric approach is robust both to the well known boundary bias problem and the curse of dimensionality problem. We derive the mean integrated squared error properties, including the rate of convergence, the uniform strong consistency and the asymptotic normality. A simulation study investigates the finite sample performance of the estimator. We find that univariate least squares cross validation, to choose the bandwidth for the estimation of the marginal densities, works well and that the estimator we propose performs very well also for data with unbounded support. Applications in the field of finance are provided.Asymptotic properties, asymmetric kernels, boundary bias, copula, curse of dimension, least squares cross validation.
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
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