A Berry-Esséen Theorem for Serial Rank Statistics

Authors: Hallin M.¹; Rifi K.²

Source: Annals of the Institute of Statistical Mathematics, Volume 49, Number 4, December 1997 , pp. 777-799(23)

Publisher: Springer

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Abstract:

Berry-Esséen bounds of the optimal O(n-1/2) order are obtained, under the null hypothesis of randomness, for serial linear rank statistics, of the form Sigma a1 (Rt)a2(Rt-k). Such statistics play an essential role in distribution-free methods for time-series analysis, where they provide nonparametric analogues to classical (Gaussian) correlogram-based methods. Berry-Esséen inequalities are established under mild conditions on the score-generating functions, allowing for normal (van der Waerden) scores. They extend to the serial case the earlier result of Does (1982, Ann. Probab., 10, 982-991) on (nonserial) linear rank statistics, and to the context of nonparametric rank-based statistics the parametric results of Taniguchi (1991, Higher Order Asymptotics for Time Series Analysis, Springer, New York) on quadratic forms of Gaussian stationary processes.

Keywords: Berry-Esséen bounds; serial rank statistics; time series

Language: English

Document Type: Regular paper

Affiliations: 1: Institut de Statistique and Département de Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP 210, Boulevard du Triomphe, B 1050 Bruxelles, Belgium 2: Ecole Normale Supérieure, Bensouda BP 34A, Fès, Morocco

Publication date: 1997-12-01