389 research outputs found
Copula Processes
We define a copula process which describes the dependencies between
arbitrarily many random variables independently of their marginal
distributions. As an example, we develop a stochastic volatility model,
Gaussian Copula Process Volatility (GCPV), to predict the latent standard
deviations of a sequence of random variables. To make predictions we use
Bayesian inference, with the Laplace approximation, and with Markov chain Monte
Carlo as an alternative. We find both methods comparable. We also find our
model can outperform GARCH on simulated and financial data. And unlike GARCH,
GCPV can easily handle missing data, incorporate covariates other than time,
and model a rich class of covariance structures.Comment: 11 pages, 1 table, 1 figure. Submitted for publication. Since last
version: minor edits and reformattin
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
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
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
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
Empirical processes indexed by estimated functions
We consider the convergence of empirical processes indexed by functions that
depend on an estimated parameter and give several alternative conditions
under which the ``estimated parameter'' can be replaced by its natural
limit uniformly in some other indexing set . In particular we
reconsider some examples treated by Ghoudi and Remillard [Asymptotic Methods in
Probability and Statistics (1998) 171--197, Fields Inst. Commun. 44 (2004)
381--406]. We recast their examples in terms of empirical process theory, and
provide an alternative general view which should be of wide applicability.Comment: Published at http://dx.doi.org/10.1214/074921707000000382 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Conditional empirical copula processes and generalized dependence measures
We study the weak convergence of conditional empirical copula processes, when
the conditioning event has a nonzero probability. The validity of several
bootstrap schemes is stated, including the exchangeable bootstrap. We define
general - possibly conditional - multivariate dependence measures and their
estimators. By applying our theoretical results, we prove the asymptotic
normality of some estimators of such dependence measures.Comment: 29 pages, 4 figure
Weak convergence of the weighted empirical beta copula process
The empirical copula has proved to be useful in the construction and
understanding of many statistical procedures related to dependence within
random vectors. The empirical beta copula is a smoothed version of the
empirical copula that enjoys better finite-sample properties. At the core lie
fundamental results on the weak convergence of the empirical copula and
empirical beta copula processes. Their scope of application can be increased by
considering weighted versions of these processes. In this paper we show weak
convergence for the weighted empirical beta copula process. The weak
convergence result for the weighted empirical beta copula process is stronger
than the one for the empirical copula and its use is more straightforward. The
simplicity of its application is illustrated for weighted Cram\'er--von Mises
tests for independence and for the estimation of the Pickands dependence
function of an extreme-value copula.Comment: 19 pages, 2 figure
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