Scanning N Uniform Distributed Points: Bounds

Abstract

In the study of bounds for scan statistic probabilities two methods have been employed. The first method utilizes the scanning process representation of the scan statistic that has been discussed in Naus (1982), Wallenstein (1980), and Wallenstein and Neff (1987). The second method is based on the order statistics representation of the scan statistics investigated in Berman and Eagleson (1985), Gates and Westcott (1984), Glaz (1989, 1992), and Krauth (1988). The class of inequalities that we will present here is known in the statistical literature as Bonferroni-type inequalities. For a thorough treatment of these inequalities and many interesting references and applications, see a recent book by Galambos and Simonelli (1996). The Bonferroni-type inequalities that we will use for developing bounds for scan statistic probabilities are usually tigher than the classical Bonferroni inequalities introduced in Bonferroni (1936). Therefore, the classical Bonferroni inequalities will not be discussed here.