Empirical processes based on pseudo-observations II: The multivariate case

Abstract

One often needs to estimate the distribution functions of a random vector E = H(X), where the H is unknown and might depend on the law of X . When H is estimated by some H,, using a sample XI,. . . , X,, the H,,(X,)'s are termed pseudeobservations. In a semiparametric context, one often wants to estimate parameters related to the law of the non-observable E . The transformed data Hn(X1), . . ., Hn(Xn) are then naturally used, introducing dependence. Classical techniques do not apply and hard work is needed to get the asymptotic behaviour of estimators and empirical processes.The aim of this paper is to give a unified treatment of inference procedures based on pseudeobservations in the multivariate setting. Exaniples of applications are given.

Rank-Based Extensions of the Brock, Dechert, and Scheinkman Test

“…Assumption I is simple and realistic. Assumption II looks more intricate, but in practice, most empirical processes based on pseudo-observations have a limit of the form specified there (see, e.g., Berkes and Horváth 2003;Ghoudi and Rémillard 2004;and references therein). The only additional restriction imposed by Assumption II is the functional independence of the term β k (x 1 , .…”

“…Consider the extended ARCH(p) model defined by Y i = X i + μ in which X i is defined as in Example 1. Using techniques from Ghoudi and Rémillard (2004), we can show that if…”

A note on bootstrap approximations for the empirical copula process

“…The asymptotic behavior of C n was studied in several papers, including Gänssler and Stute (1987), Ghoudi and Rémillard (2004) …”

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“…and letting Π denotes the d-dimensional independence copula, it then follows from [18] that under the hypothesis of independence, …”