Maxima in hypercubes

Authors:
Zhi-Dong Bai;Luc Devroye;Hsien-Kuei Hwang;Tsung-Hsi Tsai
Affiliations:
College of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, People's Republic of China and Department of Statistics & Applied Probability, National University of Singapor ...;School of Computer Science, McGill University, Montreal, Canada;Institute of Statistical Science, Academia Sinica, Taipei, Taiwan;Institute of Statistical Science, Academia Sinica, Taipei, Taiwan
Venue:
Random Structures & Algorithms
Year:
2005

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Abstract

We derive a Berry-Esseen bound, essentially of the order of the square of the standard deviation, for the number of maxima in random samples from (0, 1)d. The bound is, although not optimal, the first of its kind for the number of maxima in dimensions higher than two. The proof uses Poisson processes and Stein's method. We also propose a new method for computing the variance and derive an asymptotic expansion. The methods of proof we propose are of some generality and applicable to other regions such as d-dimensional simplex.