16,749 research outputs found
The Empirical Beta Copula
Given a sample from a multivariate distribution , the uniform random
variates generated independently and rearranged in the order specified by the
componentwise ranks of the original sample look like a sample from the copula
of . This idea can be regarded as a variant on Baker's [J. Multivariate
Anal. 99 (2008) 2312--2327] copula construction and leads to the definition of
the empirical beta copula. The latter turns out to be a particular case of the
empirical Bernstein copula, the degrees of all Bernstein polynomials being
equal to the sample size.
Necessary and sufficient conditions are given for a Bernstein polynomial to
be a copula. These imply that the empirical beta copula is a genuine copula.
Furthermore, the empirical process based on the empirical Bernstein copula is
shown to be asymptotically the same as the ordinary empirical copula process
under assumptions which are significantly weaker than those given in Janssen,
Swanepoel and Veraverbeke [J. Stat. Plan. Infer. 142 (2012) 1189--1197].
A Monte Carlo simulation study shows that the empirical beta copula
outperforms the empirical copula and the empirical checkerboard copula in terms
of both bias and variance. Compared with the empirical Bernstein copula with
the smoothing rate suggested by Janssen et al., its finite-sample performance
is still significantly better in several cases, especially in terms of bias.Comment: 23 pages, 3 figure
Approximate Bayesian inference in semiparametric copula models
We describe a simple method for making inference on a functional of a
multivariate distribution. The method is based on a copula representation of
the multivariate distribution and it is based on the properties of an
Approximate Bayesian Monte Carlo algorithm, where the proposed values of the
functional of interest are weighed in terms of their empirical likelihood. This
method is particularly useful when the "true" likelihood function associated
with the working model is too costly to evaluate or when the working model is
only partially specified.Comment: 27 pages, 18 figure
Estimation of Extreme Quantiles for Functions of Dependent Random Variables
We propose a new method for estimating the extreme quantiles for a function
of several dependent random variables. In contrast to the conventional approach
based on extreme value theory, we do not impose the condition that the tail of
the underlying distribution admits an approximate parametric form, and,
furthermore, our estimation makes use of the full observed data. The proposed
method is semiparametric as no parametric forms are assumed on all the marginal
distributions. But we select appropriate bivariate copulas to model the joint
dependence structure by taking the advantage of the recent development in
constructing large dimensional vine copulas. Consequently a sample quantile
resulted from a large bootstrap sample drawn from the fitted joint distribution
is taken as the estimator for the extreme quantile. This estimator is proved to
be consistent. The reliable and robust performance of the proposed method is
further illustrated by simulation.Comment: 18 pages, 2 figure
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