COMMENTS ON THE SURVEY BY BALAKRISHNAN AND ZHAO

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.

Download Full-text

ORDERING PROPERTIES OF ORDER STATISTICS FROM HETEROGENEOUS POPULATIONS: A REVIEW WITH AN EMPHASIS ON SOME RECENT DEVELOPMENTS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964813000156 ◽

2013 ◽

Vol 27 (4) ◽

pp. 403-443 ◽

Cited By ~ 72

Author(s):

N. Balakrishnan ◽

Peng Zhao

Keyword(s):

Order Statistics ◽

Negative Binomial ◽

Stochastic Orders ◽

Hazard Rates ◽

Stochastic Comparison ◽

Proportional Hazard ◽

Heterogeneous Populations ◽

Recent Developments ◽

Geometric Variables

In this paper, we review some recent results on the stochastic comparison of order statistics and related statistics from independent and heterogeneous proportional hazard rates models, gamma variables, geometric variables, and negative binomial variables. We highlight the close connections that exist between some classical stochastic orders and majorization-type orders.

Download Full-text

COMMENTS ON “ORDERING PROPERTIES OF ORDER STATISTICS FROM HETEROGENEOUS POPULATIONS”

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964813000181 ◽

2013 ◽

Vol 27 (4) ◽

pp. 455-462

Author(s):

Xiaohu Li ◽

Yinping You

Keyword(s):

Order Statistics ◽

Gamma Distribution ◽

Negative Binomial ◽

Geometric Distribution ◽

Stochastic Order ◽

Hazard Rates ◽

Stochastic Comparison ◽

Proportional Hazard ◽

Binomial Distributions ◽

Majorization Order

Balakrishnan and Zhao does an excellent job in this issue at reviewing the recent advances on stochastic comparison between order statistics from independent and heterogeneous observations with proportional hazard rates, gamma distribution, geometric distribution, and negative binomial distributions, the relation between various stochastic order and majorization order of concerned heterogeneous parameters is highlighted. Some examples are presented to illustrate main results while pointing out the potential direction for further discussion.

Download Full-text

ON STOCHASTIC COMPARISONS OF ORDER STATISTICS FROM HETEROGENEOUS EXPONENTIAL SAMPLES

Probability in the Engineering and Informational Sciences ◽

10.1017/s026996481900041x ◽

2019 ◽

pp. 1-6

Author(s):

Yaming Yu

Keyword(s):

Order Statistics ◽

Order Statistic ◽

Open Problem ◽

Hazard Rate ◽

Random Variables ◽

Stochastic Comparisons ◽

Heterogeneous Sample ◽

Exponential Random Variables ◽

Hazard Rate Ordering

Abstract We show that the kth order statistic from a heterogeneous sample of n ≥ k exponential random variables is larger than that from a homogeneous exponential sample in the sense of star ordering, as conjectured by Xu and Balakrishnan [14]. As a consequence, we establish hazard rate ordering for order statistics between heterogeneous and homogeneous exponential samples, resolving an open problem of Pǎltǎnea [11]. Extensions to general spacings are also presented.

Download Full-text

ORDER STATISTICS FROM HETEROGENOUS NEGATIVE BINOMIAL RANDOM VARIABLES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964811000118 ◽

2011 ◽

Vol 25 (4) ◽

pp. 435-448 ◽

Cited By ~ 5

Author(s):

Maochao Xu ◽

Taizhong Hu

Keyword(s):

Order Statistics ◽

Negative Binomial ◽

Geometric Distribution ◽

Stochastic Order ◽

Sufficient Conditions ◽

Random Variables ◽

Usual Stochastic Order ◽

Multivariate Stochastic Order ◽

Special Case ◽

Binomial Random Variables

In this article, we study the order statistics from heterogenous negative binomial random variables. Sufficient conditions are provided for comparing the extreme order statistics according to the usual stochastic order. For the special case of geometric distribution, a sufficient condition is established for comparing order statistics in the sense of multivariate stochastic order. Applications in the Poisson–Gamma shock model and redundant systems have been described as well.

Download Full-text

Stochastic comparisons of order statistics under multivariate imperfect repair

Journal of Applied Probability ◽

10.2307/3215273 ◽

1996 ◽

Vol 33 (1) ◽

pp. 156-163 ◽

Cited By ~ 5

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.

Download Full-text