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

2013 ◽  
Vol 27 (4) ◽  
pp. 455-462
Author(s):  
Xiaohu Li ◽  
Yinping You

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.

2013 ◽  
Vol 27 (4) ◽  
pp. 403-443 ◽  
Author(s):  
N. Balakrishnan ◽  
Peng Zhao

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.


2013 ◽  
Vol 27 (4) ◽  
pp. 445-449 ◽  
Author(s):  
Moshe Shaked

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.


2011 ◽  
Vol 25 (4) ◽  
pp. 435-448 ◽  
Author(s):  
Maochao Xu ◽  
Taizhong Hu

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.


2017 ◽  
Vol 33 (1) ◽  
pp. 28-49
Author(s):  
Narayanaswamy Balakrishnan ◽  
Jianbin Chen ◽  
Yiying Zhang ◽  
Peng Zhao

In this paper, we discuss the ordering properties of sample ranges arising from multiple-outlier exponential and proportional hazard rate (PHR) models. The purpose of this paper is twofold. First, sufficient conditions on the parameter vectors are provided for the reversed hazard rate order and the usual stochastic order between the sample ranges arising from multiple-outlier exponential models with common sample size. Next, stochastic comparisons are separately carried out for sample ranges arising from multiple-outlier exponential and PHR models with different sample sizes as well as different hazard rates. Some numerical examples are also presented to illustrate the results established here.


2008 ◽  
Vol 22 (3) ◽  
pp. 473-474 ◽  
Author(s):  
Subhash Kochar ◽  
Maochao Xu

In our article [3] we have found a gap in the middle of the proof of Theorem 3.2. Therefore, we do not know whether Theorem 3.2 is true for the reverse hazard rate order. However, we could prove the following weaker result for the stochastic order.


2021 ◽  
Vol 53 (1) ◽  
pp. 107-132
Author(s):  
Tomasz Rychlik ◽  
Fabio Spizzichino

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.


Author(s):  
Sameen Naqvi ◽  
Weiyong Ding ◽  
Peng Zhao

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.


2001 ◽  
Vol 16 (07) ◽  
pp. 1227-1235 ◽  
Author(s):  
C. B. YANG ◽  
X. CAI

The influence of pure statistical fluctuations on K/π ratio is investigated in an event-by-event way. Poisson and the modified negative binomial distributions are used as the multiplicity distributions since they both have statistical background. It is shown that the distributions of the ratio in these cases are Gaussian, and the mean and relative variance are given analytically.


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