On the minimization of multinomial tails and the Gupta–Nagel conjecture

Abstract

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 Gupta and Nagel (Sankhya Ser. B 29 (1967) 1) (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.

A restricted multinomial hybrid selection procedure

“…Appropriate parameters can sometimes be found analytically but often require either computationally intensive exact expansions or numerical simulations. Proofs of LFCs tend to be much more complicated for multinomial selection problems than for other types of selection procedures because the y i 's are not independent; see, for example, Kesten and Morse [1959] and Gastaldi [2005]. At the time of this writing, a proof of the LFC for M BG is not yet available, although Bechhofer and Goldsman [1986] conjecture that the slippage configuration is the LFC.…”