Gumbel’s Identity, Binomial Moments, and Bonferroni Sums

2012 ◽  
Vol 80 (2) ◽  
pp. 269-292 ◽  
Author(s):  
Fred M. Hoppe ◽  
Eugene Seneta
Keyword(s):  
Binomial Moments

scholarly journals Rank-Metric Codes, Generalized Binomial Moments and their Zeta Functions

2020 ◽  
Vol 604 ◽  
pp. 92-128
Author(s):  
Eimear Byrne ◽  
Giuseppe Cotardo ◽  
Alberto Ravagnani
Keyword(s):  
Zeta Functions ◽  
Rank Metric ◽  
Rank Metric Codes ◽  
Binomial Moments

On explicit and Fréchet-optimal lower bounds

2002 ◽  
Vol 39 (1) ◽  
pp. 81-90 ◽  
Author(s):  
Eugene Seneta ◽  
John Tuhao Chen
Keyword(s):  
Lower Bounds ◽  
Final Section ◽  
Sufficient Condition ◽  
Multiplicative Structure ◽  
Bivariate Case ◽  
Optimal Bounds ◽  
Explicit Bounds ◽  
Binomial Moments

Ease of computation of Fréchet-optimal lower bounds, given numerical values of the binomial moments Sij, i, j = 1, 2, is demonstrated. A sufficient condition is given for an explicit bivariate bound of Dawson-Sankoff structure to be Fréchet optimal. An example demonstrates that in the bivariate case even the multiplicative structure of the Sij does not guarantee a Dawson-Sankoff structure for Fréchet-optimal bounds. A final section is used to illuminate the nature of Fréchet optimality by using generalized explicit bounds. This note is a sequel to both Chen and Seneta (1995) and Chen (1998).

Multivariate Bonferroni-type lower bounds

1996 ◽  
Vol 33 (3) ◽  
pp. 729-740 ◽  
Author(s):  
Tuhao Chen ◽  
E. Seneta
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Indicator Functions ◽  
Binomial Moments

We derive multivariate Sobel–Uppuluri–Galambos-type lower bounds for the probability that at least a1 and at least a2, and for the probability that exactly a1 and a2, out of n and N events, occur. The lower bound presented here reduces to a sharper bound than that of Galambos and Lee (1992). Our approach is by way of indicator functions and bivariate binomial moments. A new concept, marginal Bonferroni summation, is introduced in this paper.

scholarly journals Probability distributions with binomial moments

2014 ◽  
Vol 17 (02) ◽  
pp. 1450014 ◽  
Author(s):  
Wojciech Młotkowski ◽  
Karol A. Penson
Keyword(s):  
Probability Distributions ◽  
Elementary Function ◽  
Positive Definite ◽  
Probability Measures ◽  
Bernoulli Distribution ◽  
Absolutely Continuous ◽  
Free Convolution ◽  
Meijer G Function ◽  
Convolution Powers ◽  
Binomial Moments

We prove that the binomial sequence [Formula: see text] is positive definite if and only if either p ≥ 1, -1 ≤ r ≤ p - 1 or p ≤ 0, p - 1 ≤ r ≤ 0 and that the Raney sequence [Formula: see text] is positive definite if and only if either p ≥ 1, 0 ≤ r ≤ p or p ≤ 0, p - 1 ≤ r ≤ 0 or else r = 0. The corresponding probability measures are denoted by ν(p, r) and μ(p, r) respectively. We prove that if p > 1 is rational and -1 < r ≤ p - 1 then the measure ν(p, r) is absolutely continuous and its density Vp,r(x) can be represented as Meijer G-function. In some cases Vp,r is an elementary function. We show that for p > 1 the measures ν(p,-1) and ν(p,0) are certain free convolution powers of the Bernoulli distribution.

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