ORDER STATISTICS FROM HETEROGENOUS NEGATIVE BINOMIAL RANDOM VARIABLES

2011 ◽  
Vol 25 (4) ◽  
pp. 435-448 ◽  
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.

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.

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.

Comparing the Variability of Random Variables and Point Processes

1998 ◽  
Vol 12 (4) ◽  
pp. 425-444 ◽  
Author(s):  
Mark Brown ◽  
J. George Shanthikumar
Keyword(s):  
Point Processes ◽  
Residual Life ◽  
Stochastic Order ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Rate Function ◽  
Poisson Processes ◽  
Mean Residual Life ◽  
Event Time ◽  
Usual Stochastic Order

In this paper we compare the variance of functions of random variables and functionals of point processes. Specifically we give sufficient conditions on two random variables X and Y under which the variances var f(X) and var f(Y) of the function f of these random variables can be compared. For example we show that if X is smaller than Y in the shifted-up mean residual life and in the usual stochastic order, then var f(X) ≤ var f(y) for all increasing convex functions f, whenever these variances are well defined. In the context of point processes we compare the variances var Φ(M) and var Φ(N) of the functional Φ of two point processes M = {M(t), t ≥ 0} and N = {N(t), t ≥ 0}. We provide sufficient conditions under which these variances can be compared. Specifically we consider comparisons between (i) two renewal processes and between (ii) two (homogeneous or nonhomogeneous) Poisson processes. For example we show that for any nonhomogeneous Poisson process N with a rate function bounded from above by λ and from below by μ and two homogeneous Poisson processes L and M with rates λ and μ, respectively, var Φ (L) ≤ var Φ (N) ≤ var Φ (M) for any functional Φ that is increasing directionally convex in the event times, whenever these variances are well defined. This, for example, implies that if Tnis the nth event time of N, then n/λ2 ≤ var (Tn) ≤ n/μ2. Some applications of these results are given.

Optimal allocation of a coherent system with statistical dependent subsystems

2021 ◽  
pp. 1-20
Author(s):  
Bin Lu ◽  
Jiandong Zhang ◽  
Rongfang Yan
Keyword(s):  
Hazard Rate ◽  
Optimal Allocation ◽  
Stochastic Order ◽  
Sufficient Conditions ◽  
Parallel Systems ◽  
Coherent System ◽  
Allocation Policy ◽  
Usual Stochastic Order ◽  
Series Systems ◽  
Parallel Series

Abstract This paper studies the optimal allocation policy of a coherent system with independent heterogeneous components and dependent subsystems, the systems are assumed to consist of two groups of components whose lifetimes follow proportional hazard (PH) or proportional reversed hazard (PRH) models. We investigate the optimal allocation strategy by finding out the number $k$ of components coming from Group A in the up-series system. First, some sufficient conditions are provided in the sense of the usual stochastic order to compare the lifetimes of two-parallel–series systems with dependent subsystems, and we obtain the hazard rate and reversed hazard rate orders when two subsystems have independent lifetimes. Second, similar results are also obtained for two-series–parallel systems under certain conditions. Finally, we generalize the corresponding results to parallel–series and series–parallel systems with multiple subsystems in the viewpoint of the minimal path and the minimal cut sets, respectively. Some numerical examples are presented to illustrate the theoretical findings.

Ordering comparison of negative binomial random variables with their mixtures

2008 ◽  
Vol 78 (14) ◽  
pp. 2234-2239 ◽  
Author(s):  
Mohammad Hossein Alamatsaz ◽  
Somayyeh Abbasi
Keyword(s):  
Negative Binomial ◽  
Random Variables ◽  
Binomial Random Variables

The stochastic precedence ordering with applications in sampling and testing

2004 ◽  
Vol 41 (01) ◽  
pp. 73-82 ◽  
Author(s):  
Philip J. Boland ◽  
Harshinder Singh ◽  
Bojan Cukic
Keyword(s):  
Random Sampling ◽  
Stochastic Order ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Simple Random Sampling ◽  
Order Type ◽  
Bernoulli Random Variables ◽  
Binomial Distributions ◽  
Necessary And Sufficient ◽  
Stochastic Precedence

Stratified and simple random sampling (or testing) are two common methods used to investigate the number or proportion of items in a population with a particular attribute. Although it is known that cost factors and information about the strata in the population are often crucial in deciding whether to use stratified or simple random sampling in a given situation, the stochastic precedence ordering for random variables can also provide the basis for an interesting criteria under which these methods may be compared. It may be particularly relevant when we are trying to find as many special items as possible in a population (for example individuals with a disease in a country). Properties of this total stochastic order on the class of random variables are discussed, and necessary and sufficient conditions are established which allow the comparison of the number of items of interest found in stratified random sampling with the number found in simple random sampling in the stochastic precedence order. These conditions are compared with other results established on stratified and simple random sampling (testing) using different stochastic-order-type criteria, and applications are given for the comparison of sums of Bernoulli random variables and binomial distributions.

STOCHASTIC COMPARISONS OF LARGEST ORDER STATISTICS FROM MULTIPLE-OUTLIER EXPONENTIAL MODELS

2012 ◽  
Vol 26 (2) ◽  
pp. 159-182 ◽  
Author(s):  
Peng Zhao ◽  
N. Balakrishnan
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Likelihood Ratio Order ◽  
Stochastic Comparisons ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Exponential Models ◽  
Usual Stochastic Order

In this paper, we carry out stochastic comparisons of largest order statistics from multiple-outlier exponential models according to the likelihood ratio order (reversed hazard rate order) and the hazard rate order (usual stochastic order). It is proved, among others, that the weak majorization order between the two hazard rate vectors is equivalent to the likelihood ratio order (reversed hazard rate order) between largest order statistics, and that the p-larger order between the two hazard rate vectors is equivalent to the hazard rate order (usual stochastic order) between largest order statistics. We also extend these results to the proportional hazard rate models. The results established here strengthen and generalize some of the results known in the literature.

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