Stochastic comparisons of order statistics under multivariate imperfect repair

1996 ◽  
Vol 33 (01) ◽  
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.

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.

COMMENTS ON THE SURVEY BY BALAKRISHNAN AND ZHAO

2013 ◽  
Vol 27 (4) ◽  
pp. 445-449 ◽  
Author(s):  
Moshe Shaked
Keyword(s):  
Order Statistics ◽  
Negative Binomial ◽  
Random Variables ◽  
Hazard Rates ◽  
Proportional Hazard ◽  
Stochastic Comparisons ◽  
Exponential Random Variables ◽  
Binomial Random Variables

N. Balakrishnan and Peng Zhao have prepared an outstanding survey of recent results that stochastically compare various order statistics and some ranges based on two collections of independent heterogeneous random variables. Their survey focuses on results for heterogeneous exponential random variables and their extensions to random variables with proportional hazard rates. In addition, some results that stochastically compare order statistics based on heterogeneous gamma, Weibull, geometric, and negative binomial random variables are also given. In particular, the authors of have listed some stochastic comparisons that are based on one heterogeneous collection of random variables, and one homogeneous collection of random variables. Personally, I find these types of comparisons to be quite fascinating. Balakrishnan and Zhao have done a thorough job of listing all the known results of this kind.

Stochastic ordering of order statistics

1991 ◽  
Vol 28 (2) ◽  
pp. 278-286 ◽  
Author(s):  
A. D. Barbour ◽  
T. Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Order Relation ◽  
Stochastic Ordering ◽  
Rate Function ◽  
Independent Random Variables ◽  
Failure Rate Function ◽  
Exponential Random Variables ◽  
The Law ◽  
Monotone Coupling

If Xi, i = 1, ···, n are independent exponential random variables with parameters λ1, · ··, λ n, and if Yi, i = 1, ···, n are independent exponential random variables with common parameter equal to (λ1 + · ·· + λ n)/n, then there is a monotone coupling of the order statistics X(1), · ··, X(n) and Y(1), · ··, Y(n); that is, it is possible to construct on a common probability space random variables X′i, Y′i, i = 1, ···, n, such that for each i, Y′(i)≦X′(i) a.s., where the law of the X′i (respectively, the Y′i) is the same as the law of the Xi (respectively, the Yi.) This result is due to Proschan and Sethuraman, and independently to Ball. We shall here prove an extension to a more general class of distributions for which the failure rate function r(x) is decreasing, and xr(x) is increasing. This very strong order relation allows comparison of properties of epidemic processes where rates of infection are not uniform with the corresponding properties for the homogeneous case. We further prove that for a sequence Zi, i = 1, ···, n of independent random variables whose failure rates at any time add to 1, the order statistics are stochastically larger than the order statistics of a sample of n independent exponential random variables of mean n, but that the strong monotone coupling referred to above is impossible in general.

A limit theorem with applications in order statistics

1974 ◽  
Vol 11 (1) ◽  
pp. 219-222 ◽  
Author(s):  
János Galambos
Keyword(s):  
Limit Theorem ◽  
Order Statistics ◽  
Limit Distribution ◽  
Probability Space ◽  
Random Variables ◽  
Analytic Method ◽  
Sufficient Condition ◽  
Dependent Random Variables ◽  
Special Case ◽  
Infinite Sequences

Let A1, A2, ···, An be events on a given probability space and let Br, n be the event that exactly r of the A's occur. Let further Sk (n) be the kth binomial moment of the number of the A's which occur. A sufficient condition is given for the existence of lim P (Br,n), as n→ + ∞, in terms of limits of the Sk(n)'s and a formula is given for the limit above. This formula for the limit is similar to the sieve theorem of Takács (1967) for infinite sequences of events and in the proof we make use of Takács's analytic method. The result is immediately applicable to the limit distribution of the maximum of (dependent) random variables X1, X2, ···, Xn by choosing Aj = {Xj ≧ x}. Our main theorem is reformulated for this special case and an example is given for illustration.

Stochastic ordering of order statistics

1991 ◽  
Vol 28 (02) ◽  
pp. 278-286 ◽  
Author(s):  
A. D. Barbour ◽  
T. Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Order Relation ◽  
Stochastic Ordering ◽  
Rate Function ◽  
Independent Random Variables ◽  
Failure Rate Function ◽  
Exponential Random Variables ◽  
The Law ◽  
Monotone Coupling

If Xi , i = 1, ···, n are independent exponential random variables with parameters λ 1, · ··, λ n , and if Yi, i = 1, ···, n are independent exponential random variables with common parameter equal to (λ1 + · ·· + λ n )/n, then there is a monotone coupling of the order statistics X (1), · ··, X (n) and Y (1), · ··, Y (n); that is, it is possible to construct on a common probability space random variables X′ i , Y′ i , i = 1, ···, n, such that for each i, Y ′ (i)≦X ′ (i) a.s., where the law of the X ′ i (respectively, the Y ′ i ) is the same as the law of the Xi (respectively, the Yi.) This result is due to Proschan and Sethuraman, and independently to Ball. We shall here prove an extension to a more general class of distributions for which the failure rate function r(x) is decreasing, and xr(x) is increasing. This very strong order relation allows comparison of properties of epidemic processes where rates of infection are not uniform with the corresponding properties for the homogeneous case. We further prove that for a sequence Zi, i = 1, ···, n of independent random variables whose failure rates at any time add to 1, the order statistics are stochastically larger than the order statistics of a sample of n independent exponential random variables of mean n, but that the strong monotone coupling referred to above is impossible in general.

SOME NEW RESULTS ON ORDERING OF SIMPLE SPACINGS OF GENERALIZED ORDER STATISTICS

2010 ◽  
Vol 25 (1) ◽  
pp. 71-81 ◽  
Author(s):  
Hongmei Xie ◽  
Weiwei Zhuang
Keyword(s):  
Order Statistics ◽  
Residual Life ◽  
Random Variables ◽  
Generalized Order Statistics ◽  
Mean Residual Life ◽  
Unified Approach ◽  
Stochastic Comparisons ◽  
The Mean ◽  
Generalized Order

The concept of generalized order statistics was introduced as a unified approach to a variety of models of ordered random variables. The purpose of this article is to establish several stochastic comparisons of simple spacings in the mean residual life and the excess wealth orders under the more general assumptions on the parameters of the models.

A limit theorem with applications in order statistics

1974 ◽  
Vol 11 (01) ◽  
pp. 219-222
Author(s):  
János Galambos
Keyword(s):  
Limit Theorem ◽  
Order Statistics ◽  
Limit Distribution ◽  
Probability Space ◽  
Random Variables ◽  
Analytic Method ◽  
Sufficient Condition ◽  
Dependent Random Variables ◽  
Special Case ◽  
Infinite Sequences

Let A 1, A 2, ···, An be events on a given probability space and let Br, n be the event that exactly r of the A's occur. Let further Sk (n) be the kth binomial moment of the number of the A's which occur. A sufficient condition is given for the existence of lim P (Br,n ), as n→ + ∞, in terms of limits of the Sk (n)'s and a formula is given for the limit above. This formula for the limit is similar to the sieve theorem of Takács (1967) for infinite sequences of events and in the proof we make use of Takács's analytic method. The result is immediately applicable to the limit distribution of the maximum of (dependent) random variables X 1, X 2, ···, Xn by choosing Aj = {Xj ≧ x}. Our main theorem is reformulated for this special case and an example is given for illustration.

STOCHASTIC PROPERTIES OF p-SPACINGS OF GENERALIZED ORDER STATISTICS

2005 ◽  
Vol 19 (2) ◽  
pp. 257-276 ◽  
Author(s):  
Taizhong Hu ◽  
Weiwei Zhuang
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Random Variables ◽  
Generalized Order Statistics ◽  
Unified Approach ◽  
Stochastic Comparisons ◽  
Stochastic Properties ◽  
Generalized Order

The concept of generalized order statistics was introduced as a unified approach to a variety of models of ordered random variables. The purpose of this article is to investigate the conditions on the parameters that enable one to establish several stochastic comparisons of general p-spacings for a subclass of generalized order statistics in the likelihood ratio and the hazard rate orders. Preservation properties of the logconvexity and logconcavity of p-spacings are also given.

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