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