Abstract
An elementary “majorant-minorant method” to construct the most stringent Bonferroni-type inequalities is presented. These are essentially Chebyshev-type inequalities for discrete probability distributions on the set {0, 1,..., n}, where n is the number of concerned events, and polynomials with specific properties on the set lead to the inequalities. All the known results are proved easily by this method. Further, the inequalities in terms of all the lower moments are completely solved by the method. As examples, the most stringent new inequalities of degrees three and four are obtained. Simpler expressions of Mărgăritescu's inequality (1987, Stud. Cerc. Mat., 39, 246–251), improving Galambos' inequality, are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alt, F. B. (1982). Bonferroni inequalities and intervals, Encyclopedia of Statistical Sciences (eds. S. Kotz and N. L. Johnson), Vol. 1, 294–300, Wiley, New York.
Galambos, J. (1975). Methods for proving Bonferroni type inequalities, J. London Math. Soc. (2), 9, 561–564.
Galambos, J. (1977). Bonferroni inequalities, Ann. Probab., 5, 577–581.
Galambos, J. (1984). Order statistics, Handbook of Statistics, Vol. 4, Nonparametric Methods (eds. P. R. Krishnaiah and P. K. Sen), 359–382, North Holland, Amsterdam.
Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statics, 2nd ed., Robert E. Krieger Publ., Malabar, Florida.
Hailperin, T. (1965). Best possible inequalities for the probability of a logical function of events, Amer. Math. Monthly, 72, 343–359.
Hunter, D. (1976). An upper bound for the probability of a union, J. Appl. Probab., 13, 597–603.
Isii, K. (1964). Inequalities of the types of Chebyshev and Cramér-Rao and mathematical programming, Ann. Inst. Statist. Math., 16, 277–293.
Jordan, C. (1960). Calculus of Finite Difference, Chelsea, New York.
Keilson, J. and Gerber, H. (1971). Some results for discrete unimodality, J. Amer. Statist. Assoc., 66, 386–389.
Knuth, D. (1975). Fundamental Algorithms, The Art of Computer Programming, Vol. 1, 2nd ed., Addison-Wesley, Reading, Massachusetts.
Kounias, S. and Marin, J. (1976). Best linear Bonferroni bounds, SIAM J. Appl. Math., 30, 307–323.
Kwerel, S. M. (1975a). Most stringent bounds on aggregated probabilities of partially specified dependent probability systems, J. Amer. Statist. Assoc., 70, 472–479.
Kwerel, S. M. (1975b). Bounds on the probability of the union and intersection of m events, Adv. in Appl. Probab., 7, 431–448.
Kwerel, S. M. (1975c). Most stringent bounds on the probability of the union and intersection of m events for systems partially specified by S 1, S 2, ..., S k , 2≤k≤m, J. Appl. Probab., 12, 612–619.
Loève, M. (1942). Sur les systèmes d'événements. Ann. Univ. Lyon, A, 5, 55–74.
Loève, M. (1963). Probability Theory, 2nd ed., Van Nostrand, Princeton, New Jersey.
Mărgăritescu, E. (1987). On some Bonferroni inequalities, Stud. Cerc. Mat., 39, 246–251.
Móri, T. F. and Székely, G. J. (1985). A note on the background of several Bonferroni-Galambos-type inequalities, J. Appl. Probab., 22, 836–843.
Platz, O. (1985). A sharp upper probability bound for the occurrence of at least m out of n events, J. Appl. Probab., 12, 978–981.
Recsei, E. and Seneta, E. (1987). Bonferroni-type inequalities, Adv. in Appl. Probab., 19, 508–511.
Rényi, A. (1961). A general method for proving theorems in probability theory and some applications, MTA III. Oszt. Közl, 11, 79–105. (English translation (1976). Selected Papers of A. Rényi, Vol. 2, 581–602, Akadémiai Kiadó, Budapest.)
Riordan, J. (1968). Combinatorial Identities, Wiley, New York.
Samuels, S. M. and Studden, W. J. (1989). Bonferroni-type probability bounds as an application of the theory of Tchebycheff systems, Probability, Statistics, and Mathematics, Papers in Honor of Samuel Karlin (eds. T. W. Anderson, K. B. Athreya and D. L. Iglehart), Academic Press, Boston, Massachussets.
Sathe, Y. S., Pradhan, M. and Shah, S. P. (1980). Inequalities for the probability of the occurrence of at least m out of n events, J. Appl. Probab., 17, 1127–1132.
Schwager, S. J. (1984). Bonferroni sometimes loses, Amer. Statist., 38, 192–197.
Sibuya, M. (1991). An identity for sums of binomial coefficients, SIAM Rev. (in print).
Simes, R. J. (1986). An improved Bonferroni procedure for multiple tests of significance, Biometrika, 73, 751–754.
Takeuchi, K. and Takemura, A. (1987). On sum of 0–1 random variables I. Univariate case, Ann. Inst. Statist. Math., 39, 85–102.
Walker, A. M. (1981). On the classical Bonferroni inequalities and the corresponding Galambos inequalities, J. Appl. Probab., 18, 757–763.
Worsley, K. J. (1982). An improved Bonferroni inequality and applications, Biometrika, 69, 297–302.
Author information
Authors and Affiliations
About this article
Cite this article
Sibuya, M. Bonferroni-type inequalities; Chebyshev-type inequalities for the distributions on [0, n]. Ann Inst Stat Math 43, 261–285 (1991). https://doi.org/10.1007/BF00118635
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00118635