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

2013 ◽  
Vol 27 (4) ◽  
pp. 403-443 ◽  
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.

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

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.

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.

EQUIVALENT CHARACTERIZATIONS ON ORDERINGS OF ORDER STATISTICS AND SAMPLE RANGES

2010 ◽  
Vol 24 (2) ◽  
pp. 245-262 ◽  
Author(s):  
Tiantian Mao ◽  
Taizhong Hu
Keyword(s):  
Order Statistics ◽  
Exponential Distribution ◽  
Geometric Distribution ◽  
Random Variables ◽  
The Other ◽  
Stochastic Orders ◽  
Exponential Random Variables ◽  
Discrete Counterpart ◽  
Similarities And Differences ◽  
Geometric Variables

The purpose of this article is to present several equivalent characterizations of comparing the largest-order statistics and sample ranges of two sets of n independent exponential random variables with respect to different stochastic orders, where the random variables in one set are heterogeneous and the random variables in the other set are identically distributed. The main results complement and extend several known results in the literature. The geometric distribution can be regarded as the discrete counterpart of the exponential distribution. We also study the orderings of the largest-order statistics from geometric random variables and point out similarities and differences between orderings of the largest-order statistics from geometric variables and from exponential variables.

scholarly journals Stochastic Comparisons of Order Statistics and Spacings: A Review

2012 ◽  
Vol 2012 ◽  
pp. 1-47 ◽  
Author(s):  
Subhash Kochar
Keyword(s):  
Order Statistics ◽  
Hazard Rate ◽  
Proportional Hazard ◽  
Rate Model ◽  
Stochastic Comparisons ◽  
Scale Parameters ◽  
Hazard Rate Model ◽  
Recent Developments ◽  
Parent Observations ◽  
Sample Spacings

We review some of the recent developments in the area of stochastic comparisons of order statistics and sample spacings. We consider the cases when the parent observations are identically as well as nonidentically distributed. But most of the time we will be assuming that the observations are independent. The case of independent exponentials with unequal scale parameters as well as the proportional hazard rate model is discussed in detail.

COMMENTS ON: ORDERING PROPERTIES OF ORDER STATISTICS FROM HETEROGENEOUS POPULATIONS: A REVIEW WITH AN EMPHASIS ON SOME RECENT DEVELOPMENTS BY N. BALAKRISHNAN AND P. ZHAO

2013 ◽  
Vol 27 (4) ◽  
pp. 463-464
Author(s):  
Baha-Eldin Khaledi
Keyword(s):  
Order Statistics ◽  
Review Paper ◽  
Stochastic Comparisons ◽  
Heterogeneous Populations ◽  
Excellent Review ◽  
Recent Developments

I thank the authors for providing another excellent review on stochastic comparisons of order statistics after the review paper written by Kochar and Xu (2007).

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

2013 ◽  
Vol 27 (4) ◽  
pp. 451-454 ◽  
Author(s):  
Maochao Xu
Keyword(s):  
Order Statistics ◽  
Shape Parameters ◽  
Stochastic Comparisons ◽  
Heterogeneous Populations ◽  
Recent Developments

Professors Balakrishnan and Zhao have written an excellent survey on the recent developments of stochastic comparisons of order statistics, which cover almost every aspect of ordering properties of order statistics from both continuous and discrete heterogeneous populations. My discussion will be limited to the skewness of order statistics and order statistics from heterogeneous populations with different shape parameters.

Load-Sharing Reliability Models with Different Component Sensitivities to Other Components’ Working States

2021 ◽  
Vol 53 (1) ◽  
pp. 107-132
Author(s):  
Tomasz Rychlik ◽  
Fabio Spizzichino
Keyword(s):  
Order Statistics ◽  
Load Sharing ◽  
Single Component ◽  
Generalized Order Statistics ◽  
Hazard Rates ◽  
System Operation ◽  
Reliability Models ◽  
Mixtures Of Distributions ◽  
Failure Times ◽  
Generalized Order

AbstractWe study the distributions of component and system lifetimes under the time-homogeneous load-sharing model, where the multivariate conditional hazard rates of working components depend only on the set of failed components, and not on their failure moments or the time elapsed from the start of system operation. Then we analyze its time-heterogeneous extension, in which the distributions of consecutive failure times, single component lifetimes, and system lifetimes coincide with mixtures of distributions of generalized order statistics. Finally we focus on some specific forms of the time-nonhomogeneous load-sharing model.

Stochastic comparison of parallel systems with Pareto components

2021 ◽  
pp. 1-13
Author(s):  
Sameen Naqvi ◽  
Weiyong Ding ◽  
Peng Zhao
Keyword(s):  
Order Statistics ◽  
Extreme Value Theory ◽  
Pareto Distribution ◽  
Parallel Systems ◽  
Value Theory ◽  
Extreme Value ◽  
Stochastic Comparison ◽  
Shape Parameters ◽  
Scale Parameters ◽  
Dispersive Ordering

Abstract Pareto distribution is an important distribution in extreme value theory. In this paper, we consider parallel systems with Pareto components and study the effect of heterogeneity on skewness of such systems. It is shown that, when the lifetimes of components have different shape parameters, the parallel system with heterogeneous Pareto component lifetimes is more skewed than the system with independent and identically distributed Pareto components. However, for the case when the lifetimes of components have different scale parameters, the result gets reversed in the sense of star ordering. We also establish the relation between star ordering and dispersive ordering by extending the result of Deshpande and Kochar [(1983). Dispersive ordering is the same as tail ordering. Advances in Applied Probability 15(3): 686–687] from support $(0, \infty )$ to general supports $(a, \infty )$ , $a > 0$ . As a consequence, we obtain some new results on dispersion of order statistics from heterogeneous Pareto samples with respect to dispersive ordering.

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