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 Σ 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.
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Hallin, M., Rifi, K. A Berry-Esséen Theorem for Serial Rank Statistics. Annals of the Institute of Statistical Mathematics 49, 777–799 (1997). https://doi.org/10.1023/A:1003286814679
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DOI: https://doi.org/10.1023/A:1003286814679