508 research outputs found
Bounds on Integrals with Respect to Multivariate Copulas
Finding upper and lower bounds to integrals with respect to copulas is a
quite prominent problem in applied probability. In their 2014 paper, Hofer and
Iaco showed how particular two dimensional copulas are related to optimal
solutions of the two dimensional assignment problem. Using this, they managed
to approximate integrals with respect to two dimensional copulas. In this
paper, we will further illuminate this connection, extend it to d-dimensional
copulas and therefore generalize the method from Hofer and Iaco to arbitrary
dimensions. We also provide convergence statements. As an example, we consider
three dimensional dependence measures
Rank-based inference for bivariate extreme-value copulas
Consider a continuous random pair whose dependence is characterized
by an extreme-value copula with Pickands dependence function . When the
marginal distributions of and are known, several consistent estimators
of are available. Most of them are variants of the estimators due to
Pickands [Bull. Inst. Internat. Statist. 49 (1981) 859--878] and
Cap\'{e}ra\`{a}, Foug\`{e}res and Genest [Biometrika 84 (1997) 567--577]. In
this paper, rank-based versions of these estimators are proposed for the more
common case where the margins of and are unknown. Results on the limit
behavior of a class of weighted bivariate empirical processes are used to show
the consistency and asymptotic normality of these rank-based estimators. Their
finite- and large-sample performance is then compared to that of their
known-margin analogues, as well as with endpoint-corrected versions thereof.
Explicit formulas and consistent estimates for their asymptotic variances are
also given.Comment: Published in at http://dx.doi.org/10.1214/08-AOS672 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The Bivariate Normal Copula
We collect well known and less known facts about the bivariate normal
distribution and translate them into copula language. In addition, we prove a
very general formula for the bivariate normal copula, we compute Gini's gamma,
and we provide improved bounds and approximations on the diagonal.Comment: 24 page
- …