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On the minimization of multinomial tails and the Gupta-Nagel conjecture. (English) Zbl 1067.62053

Summary: This paper is primarily concerned with the open problem of minimizing the lower tail of the multinomial distribution. During the study of that specific problem, we have developed an approach which reveals itself useful for solving a general class of problems involving multinomial probabilities. Concerning the main problem, we provide a self-contained proof that the minimum of the multinomial lower tail is reached, as conjectured by S. S. Gupta and K. Nagel [Sankhyā, Ser. B 29, 1–34 (1967)] (within the framework of subset-selection problems), at the equal probability configuration, i.e., when the cell probabilities are equal to one another. We also point out some novel inequalities and general properties involving multinomial probabilities and multinomial coefficients.

MSC:

62G32 Statistics of extreme values; tail inference
62F07 Statistical ranking and selection procedures
62E10 Characterization and structure theory of statistical distributions
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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References:

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