# Lower Bounds for the Tail Probabilities of the Scan Statistic

Conference paper

## Abstract

The scan statistic is used for testing the null hypothesis of uniformity against clustering alternatives. Berman & Eagleson (1985) derived an upper bound for the tail probabilities which was improved by Krauth (1988) and Glaz (1989). Glaz (1989) also derived a lower bound, based on a result of Kwerel (1975). This article presents lower bounds for the scan statistic that are easier to compute. They are proved by the method of indicators or by a linear programming approach.

## Keywords

Lower Bound Marked Point American Statistical Association Tail Probability Linear Programming Approach
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## Bibliography

- Berman, M., Eagleson, G.K. (1985): A useful upper bound for the tail probabilities of the scan statistic when the sample size is large. J. of the American Statistical Association 80 886–889.MathSciNetCrossRefGoogle Scholar
- Boros, E., Prékopa, A. (1989): Closed form two-sided bounds for probabilities that at least
*r*and exactly*r*out of*n*events occur. Mathematics of Operations Research 14 317–342.zbMATHMathSciNetCrossRefGoogle Scholar - Dawson, D.A., Sankoff, D (1967): An inequality for probabilities. Proceedings of the American Mathematical Society 18 504–507.zbMATHMathSciNetCrossRefGoogle Scholar
- Galambos, J. (1975): Methods for proving Bonferroni type inequalities. J. of the London Mathematical Society 9 561–564.zbMATHMathSciNetCrossRefGoogle Scholar
- Glaz, J. (1989): Approximations and bounds for the distribution of the scan statistic. J. of the American Statistical Association 84 560–566.zbMATHMathSciNetCrossRefGoogle Scholar
- Hoppe, F.M. (1985): Iterating Bonferroni bounds. Statistics and Probability Letters 3 121–125.zbMATHMathSciNetCrossRefGoogle Scholar
- Kounias, S., Sotirakoglou, K. (1989): Bonferroni bounds revisited. J. of Applied Probability 26 233–241.zbMATHMathSciNetCrossRefGoogle Scholar
- Krauth, J. (1988): An improved upper bound for the tail probability of the scan statistic for testing nonrandom clustering. Proc. First Conference of the International Federation of Classification Societies. North-Holland, Amsterdam, 237–244.Google Scholar
- Kwerel, S.M. (1975a): Most stringent bounds on aggregated probabilities of partially specified dependent probability systems. J. of the American Statistical Association 70 472–479.zbMATHMathSciNetCrossRefGoogle Scholar
- Kwerel, S.M. (1975b): Bounds on the probability of the union and intersection of
*m*events. Advances in Applied Probability 7 431–448.zbMATHMathSciNetCrossRefGoogle Scholar - Mărgăritescu, E. (1988): Improved Bonferroni inequalities. Revue Roumanie de Mathémathique Pures et Appliquées 33 509–515.zbMATHGoogle Scholar
- Neff, N.D., Naus, J.I. (1980): The distribution of the size of the maximum cluster of points on a line. Vol. 6 in IMS Series of Selected Tables in Mathematical Statistics. American Mathematical Society, Providence, RI.Google Scholar
- Prékopa, A. (1988): Boole-Bonferroni inequalities and linear programming. Operations Research 36 145–162.zbMATHMathSciNetCrossRefGoogle Scholar
- Pyke, R. (1965): Spacings. J. of the Royal Statistical Society (Ser. B) 27 395–449.zbMATHMathSciNetGoogle Scholar
- Seneta, E. (1988): Degree iteration and permutation in improving Bonferroni-type bounds. Australian Journal of Statistics 30A 27–38.MathSciNetCrossRefGoogle Scholar
- Tomescu, I. (1986): Hypertrees and Bonferroni inequalities. J. Combinatorial Theory. (Ser. B) 41 209–217.zbMATHMathSciNetCrossRefGoogle Scholar
- Worsley, K.J. (1985): Bonferroni (improved) wins again. The American Statistician 39 235.Google Scholar

## Copyright information

© Springer-Verlag Berlin · Heidelberg 1991