scholarly journals ON STOCHASTIC COMPARISONS OF ORDER STATISTICS FROM HETEROGENEOUS EXPONENTIAL SAMPLES

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.

Hazard rate ordering of the largest order statistics from geometric random variables

2018 ◽  
Vol 55 (2) ◽  
pp. 652-658
Author(s):  
Bara Kim ◽  
Jeongsim Kim
Keyword(s):  
Order Statistics ◽  
Open Problem ◽  
Hazard Rate ◽  
Random Variables ◽  
Hazard Rate Order ◽  
Two Samples ◽  
Hazard Rate Ordering

Abstract Mao and Hu (2010) left an open problem about the hazard rate order between the largest order statistics from two samples of n geometric random variables. Du et al. (2012) solved this open problem when n = 2, and Wang (2015) solved for 2 ≤ n ≤ 9. In this paper we completely solve this problem for any value of n.

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.

Some new results on stochastic comparisons of parallel systems

2000 ◽  
Vol 37 (4) ◽  
pp. 1123-1128 ◽  
Author(s):  
Baha-Eldin Khaledi ◽  
Subhash Kochar
Keyword(s):  
Lower Bound ◽  
Hazard Rate ◽  
Geometric Mean ◽  
Random Variables ◽  
Parallel Systems ◽  
Rate Function ◽  
Arithmetic Mean ◽  
Hazard Rate Function ◽  
Stochastic Comparisons ◽  
Exponential Random Variables

Let X1,…,Xn be independent exponential random variables with Xi having hazard rate . Let Y1,…,Yn be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏i=1nλi)1/n, the geometric mean of the λis. Let Xn:n = max{X1,…,Xn}. It is shown that Xn:n is greater than Yn:n according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of Xn:n and an upper bound on the hazard rate function of Xn:n in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference65, 203–211), which are in terms of the arithmetic mean of the λis. Furthermore, let X1*,…,Xn∗ be another set of independent exponential random variables with Xi∗ having hazard rate λi∗, i = 1,…,n. It is proved that if (logλ1,…,logλn) weakly majorizes (logλ1∗,…,logλn∗, then Xn:n is stochastically greater than Xn:n∗.

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.

Some new results on stochastic comparisons of parallel systems

2000 ◽  
Vol 37 (04) ◽  
pp. 1123-1128 ◽  
Author(s):  
Baha-Eldin Khaledi ◽  
Subhash Kochar
Keyword(s):  
Lower Bound ◽  
Hazard Rate ◽  
Geometric Mean ◽  
Random Variables ◽  
Parallel Systems ◽  
Rate Function ◽  
Arithmetic Mean ◽  
Hazard Rate Function ◽  
Stochastic Comparisons ◽  
Exponential Random Variables

Let X 1,…,X n be independent exponential random variables with X i having hazard rate . Let Y 1,…,Y n be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏ i=1 n λ i )1/n , the geometric mean of the λis. Let X n:n = max{X 1,…,X n }. It is shown that X n:n is greater than Y n:n according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of X n:n and an upper bound on the hazard rate function of X n:n in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference 65, 203–211), which are in terms of the arithmetic mean of the λ i s. Furthermore, let X 1 *,…,X n ∗ be another set of independent exponential random variables with X i ∗ having hazard rate λ i ∗, i = 1,…,n. It is proved that if (logλ1,…,logλ n ) weakly majorizes (logλ1 ∗,…,logλ n ∗, then X n:n is stochastically greater than X n:n ∗.

COMPARISONS ON LARGEST ORDER STATISTICS FROM HETEROGENEOUS GAMMA SAMPLES

2020 ◽  
pp. 1-20
Author(s):  
Ting Zhang ◽  
Yiying Zhang ◽  
Peng Zhao
Keyword(s):  
Order Statistics ◽  
Hazard Rate ◽  
Sufficient Conditions ◽  
Stochastic Ordering ◽  
Shape Parameters ◽  
Reliability Engineering ◽  
Stochastic Comparisons ◽  
Maximum Order ◽  
Scale Parameters ◽  
Hazard Rate Ordering

This paper deals with stochastic comparisons of the largest order statistics arising from two sets of independent and heterogeneous gamma samples. It is shown that the weak supermajorization order between the vectors of scale parameters together with the weak submajorization order between the vectors of shape parameters imply the reversed hazard rate ordering between the corresponding maximum order statistics. We also establish sufficient conditions of the usual stochastic ordering in terms of the p-larger order between the vectors of scale parameters and the weak submajorization order between the vectors of shape parameters. Numerical examples and applications in auction theory and reliability engineering are provided to illustrate these results.

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.

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