A BERRY-ESSÉEN THEOREM FOR SERIAL RANK STATISTICS

MARC HALLIN^{1} AND KHALID RIFI^{2}

1Institut de Statistique and Département de Mathématique, Université Libre de Bruxelles,

Campus de la Plaine CP 210, Boulevard du Triomphe, B 1050 Bruxelles, Belgium 2Ecole Normale Supérieure, Bensouda BP 34A, Fès, Morocco

(Received February 15, 1996; revised September 19, 1996)

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 ∑ a_{1}(R_{t})a_{2}(R_{t−k}). Such statistics play an essential role in distributionfree 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.

Key words and phrases: Berry-Esséen bounds, serial rank statistics, time series.