Bivariate Binomial Moments and Bonferroni-Type Inequalities

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).

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Multivariate Bonferroni-type lower bounds

Journal of Applied Probability ◽

10.2307/3215354 ◽

1996 ◽

Vol 33 (3) ◽

pp. 729-740 ◽

Cited By ~ 6

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.

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Extreme value estimation for a function of a random sample using binomial moments scheme and Boolean functions of events

Discrete Applied Mathematics ◽

10.1016/j.dam.2016.11.010 ◽

2017 ◽

Vol 219 ◽

pp. 210-218 ◽

Cited By ~ 2

Author(s):

Jinwook Lee ◽

Jongpil Kim ◽

András Prékopa

Keyword(s):

Random Sample ◽

Boolean Functions ◽

Extreme Value ◽

Value Estimation ◽

Binomial Moments

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Probability distributions with binomial moments

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025714500143 ◽

2014 ◽

Vol 17 (02) ◽

pp. 1450014 ◽

Cited By ~ 8

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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Tandem queues with bulk arrivals, infinitely many servers and correlated service times

Journal of Applied Probability ◽

10.2307/3215191 ◽

1997 ◽

Vol 34 (1) ◽

pp. 248-257 ◽

Cited By ~ 3

Author(s):

Ushio Sumita ◽

Yasushi Masuda

Keyword(s):

Service Time ◽

Numerical Experiments ◽

Queueing System ◽

Tandem Queues ◽

Renewal Equations ◽

Occupancy Distribution ◽

Service Times ◽

Binomial Moments ◽

Counterintuitive Result

A system of GIx/G/∞ queues in tandem is considered where the service times of a customer are correlated but the service time vectors for customers are independently and identically distributed. It is shown that the binomial moments of the joint occupancy distribution can be generated by a sequence of renewal equations. The distribution of the joint occupancy level is then expressed in terms of the binomial moments. Numerical experiments for a two-station tandem queueing system demonstrate a somewhat counterintuitive result that the transient covariance of the joint occupancy level decreases as the covariance of the service times increases. It is also shown that the analysis is valid for a network of GIx/SM/∞ queues.

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