Probability inequalities via negative dependence for random variables conditioned on order statistics

1987 ◽  
Vol 34 (4) ◽  
pp. 547-554 ◽  
Author(s):  
Henry W. Block ◽  
Vanderlei Bueno ◽  
Thomas H. Savits ◽  
Moshe Shaked
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Negative Dependence ◽  
Probability Inequalities

Stochastic comparisons of order statistics under multivariate imperfect repair

1996 ◽  
Vol 33 (1) ◽  
pp. 156-163 ◽  
Author(s):  
Taizhong Hu
Keyword(s):  
Order Statistics ◽  
Probability Space ◽  
Random Variables ◽  
Stochastic Comparisons ◽  
Special Model ◽  
Imperfect Repair ◽  
Monotone Coupling ◽  
Common Probability

A monotone coupling of order statistics from two sets of independent non-negative random variables Xi, i = 1, ···, n, and Yi, i = 1, ···, n, means that there exist random variables X′i, Y′i, i = 1, ···, n, on a common probability space such that , and a.s. j = 1, ···, n, where X(1) ≦ X(2) ≦ ·· ·≦ X(n) are the order statistics of Xi, i = 1, ···, n (with the corresponding notations for the X′, Y, Y′ sample). In this paper, we study the monotone coupling of order statistics of lifetimes in two multi-unit systems under multivariate imperfect repair. Similar results for a special model due to Ross are also given.

Some remarks on probability inequalities for sums of bounded convex random variables

1975 ◽  
Vol 12 (1) ◽  
pp. 155-158 ◽  
Author(s):  
M. Goldstein
Keyword(s):  
Convex Function ◽  
Exponential Convergence ◽  
Random Variables ◽  
Upper Bounds ◽  
Independent Random Variables ◽  
Probability Inequalities ◽  
Continuous Convex Function ◽  
Bounded Convex

Let X1, X2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi, i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.

scholarly journals On the distance between convex-ordered random variables, with applications

2002 ◽  
Vol 34 (2) ◽  
pp. 349-374 ◽  
Author(s):  
Michael V. Boutsikas ◽  
Eutichia Vaggelatou
Keyword(s):  
Random Variables ◽  
Simple Approximation ◽  
Negative Dependence ◽  
Exponential Approximation ◽  
Probability Metrics ◽  
Convex Ordering ◽  
Compound Poisson ◽  
Approximation Techniques ◽  
Negatively Dependent Random Variables ◽  
Dependence Concepts

Simple approximation techniques are developed exploiting relationships between generalized convex orders and appropriate probability metrics. In particular, the distance between s-convex ordered random variables is investigated. Results connecting positive or negative dependence concepts and convex ordering are also presented. These results lead to approximations and bounds for the distributions of sums of positively or negatively dependent random variables. Applications and extensions of the main results pertaining to compound Poisson, normal and exponential approximation are provided as well.

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (01) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X 1, X 2, …, X n be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

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