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Tail behaviour of Gaussian processes with applications to the Brownian pillow
, 2001
"... In this paper we investigate the tail behaviour of a random variable which may be viewed as a functional of a zero mean Gaussian process , taking special interest in the situation where obeys the structure which is typical for limiting processes ocurring in nonparametric testing of [multivaria ..."
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Cited by 4 (3 self)
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In this paper we investigate the tail behaviour of a random variable which may be viewed as a functional of a zero mean Gaussian process , taking special interest in the situation where obeys the structure which is typical for limiting processes ocurring in nonparametric testing of [multivariate] indepencency and [multivariate] constancy over time. The tail behaviour of is described by means of a constant and a random variable which is defined on the same probability space as . The constant acts as an upper bound, and is relevant for the computation of the efficiency of test statistics converging in distribution to . The random variable acts as a lower bound, and is instrumental in deriving approximations for the upper percentage points of by simulation.
ASYMPTOTIC LOCAL EFFICIENCY OF CRAMÉR–VON MISES TESTS FOR MULTIVARIATE INDEPENDENCE
, 708
"... and Rémillard [Test 13 (2004) 335–369] have shown that powerful rank tests of multivariate independence can be based on combinations of asymptotically independent Cramér–von Mises statistics derived from a Möbius decomposition of the empirical copula process. A result on the largesample behavior of ..."
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Cited by 2 (0 self)
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and Rémillard [Test 13 (2004) 335–369] have shown that powerful rank tests of multivariate independence can be based on combinations of asymptotically independent Cramér–von Mises statistics derived from a Möbius decomposition of the empirical copula process. A result on the largesample behavior of this process under contiguous sequences of alternatives is used here to give a representation of the limiting distribution of such test statistics and to compute their relative local asymptotic efficiency. Local power curves and asymptotic relative efficiencies are compared under familiar classes of copula alternatives. 1. Introduction. In
On quadratic functionals of the Brownian sheet and related processes
, 2004
"... Motivated by asymptotic problems in the theory of empirical processes, and specifically by tests of independence, we study the law of quadratic functionals of the (weighted) Brownian sheet and of the bivariate Brownian bridge on [0, 1] . In particular: (i) we use Fubini type techniques to estab ..."
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Motivated by asymptotic problems in the theory of empirical processes, and specifically by tests of independence, we study the law of quadratic functionals of the (weighted) Brownian sheet and of the bivariate Brownian bridge on [0, 1] . In particular: (i) we use Fubini type techniques to establish identities in law with quadratic functionals of other Gaussian processes, (ii) we explicitly calculate the Laplace transform of such functionals by means of KarhunenLoeve expansions, (iii) we prove central and noncentral limit theorems in the same spirit of Peccati and Yor (2004) and Nualart and Peccati (2004). Our results extend some classical computations due to P. Levy (1950), as well as the formulae recently obtained by Deheuvels and Martynov (2003).