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Empirical processes based on pseudo-obervations. II: The multivariate case. (English) Zbl 1079.60024
Horvárth, Lajos (ed.) et al., Asymptotic methods in stochastics. Festschrift for Miklós Csörgő. Proceedings of the international conference, held in honour of the work of Miklós Csörgő on the occasion of his 70th birthday, Ottawa, Canada, May 23–25, 2002. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3561-0/hbk). Fields Institute Communications 44, 381-406 (2004).
[For part I see in: Asymptotic methods in probability and statistics, 171–197 (1998; Zbl 0959.62044).]
Let $$(X_1, X_2, \dots, X_n)$$ be a random vector which is stationary and ergodic. A sequence of pseudo observations based on this random vector are defined as $$(H_n (X_1 ),\dots, H_n (X_n ))$$, where $$H_n(t)$$ is an estimate of the unknown function $$H(t)$$ based on the sample $$(X_1, X_2, \dots, X_n)$$. The weak convergence of the empirical process based on the pseudo observations is derived based on some general conditions. The general results are applied to several examples including the serial copula process and the residual process in multivariate regression models. Some applications to statistics are also considered, such as the derivation of asymptotic null distribution for the test statistics in the tests of serial independence and functional hypotheses in time series.
For the entire collection see [Zbl 1054.60001].

##### MSC:
 60F05 Central limit and other weak theorems 60G10 Stationary stochastic processes 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62J05 Linear regression; mixed models 62G10 Nonparametric hypothesis testing