scholarly journals A new stochastic order based on discrete Laplace transform and some ordering results of the order statistics

2021 ◽  
pp. 1-18
Author(s):  
Fatemeh Gharari ◽  
Masoud Ganji
Keyword(s):  
Laplace Transform ◽  
Order Statistics ◽  
Stochastic Order

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.

A new stochastic order based upon Laplace transform with applications

2009 ◽  
Vol 139 (8) ◽  
pp. 2624-2630 ◽  
Author(s):  
Xiaohu Li ◽  
Xiaoliang Ling ◽  
Ping Li
Keyword(s):  
Laplace Transform ◽  
Stochastic Order

Weak limit for the stationary distribution of the infinite-alleles model

1979 ◽  
Vol 16 (3) ◽  
pp. 459-472
Author(s):  
Haim Avni
Keyword(s):  
Laplace Transform ◽  
Order Statistics ◽  
Frequency Spectrum ◽  
Stationary Distribution ◽  
Weak Limit ◽  
Limit Behavior ◽  
Ewens Sampling Formula ◽  
Sampling Formula ◽  
Spectrum Density ◽  
Infinite Alleles Model

The limit behavior of the stationary distribution of the infinite-alleles model is reduced to a single Laplace transform formula. Some known results, such as Ewens' sampling formula, the distribution of the order-statistics and the frequency spectrum density are shown to follow from this relation. All the results are obtained within the framework of the configuration process, without recourse to finite alleles models.In view of some recent results by Kingman (1977), the results apply wherever the Ewens' sampling formula is valid.

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.

STOCHASTIC ORDERING OF A CLASS OF SYMMETRIC DISTRIBUTIONS

2009 ◽  
Vol 23 (4) ◽  
pp. 583-595
Author(s):  
Weiwei Zhuang ◽  
Taizhong Hu
Keyword(s):  
Order Statistics ◽  
Necessary Conditions ◽  
Stochastic Order ◽  
Stochastic Ordering ◽  
Random Vectors ◽  
Convex Order ◽  
Symmetric Distributions ◽  
Increasing Convex Order ◽  
Usual Stochastic Order ◽  
Directionally Convex Order

In this article, we investigate the sufficient and/or necessary conditions in order to stochastically compare the order statistics and their spacing vectors of two random vectors X and Y with special symmetric distributions. The conditions are imposed on the sample ranges Xn:n–X1:n and Yn:n–Y1:n or on (X1:n, Xn:n–X1:n) and (Y1:n, Yn:n–Y1:n). In particular, we consider the multivariate usual stochastic order, the convex order, the increasing convex order, and the directionally convex order. Several examples are also given to illustrate the power of the main results.

scholarly journals Inverse Stochastic Dominance, Majorization, and Mean Order Statistics

2010 ◽  
Vol 47 (1) ◽  
pp. 277-292 ◽  
Author(s):  
Jesús De La Cal ◽  
Javier Cárcamo
Keyword(s):  
Social Inequality ◽  
Order Statistics ◽  
Stochastic Dominance ◽  
Natural Extension ◽  
Stochastic Order ◽  
Stochastic Orderings ◽  
Natural Extensions ◽  
Weak Majorization ◽  
Two Populations ◽  
Inverse Stochastic Dominance

The inverse stochastic dominance of degree r is a stochastic order of interest in several branches of economics. We discuss it in depth, the central point being the characterization in terms of the weak r-majorization of the vectors of expected order statistics. The weak r-majorization (a notion introduced in the paper) is a natural extension of the classical (reverse) weak majorization of Hardy, Littlewood and Pòlya. This work also shows the equivalence between the continuous majorization (of higher order) and the discrete r-majorization. In particular, our results make it clear that the cases r = 1, 2 differ substantially from those with r ≥ 3, a fact observed earlier by Muliere and Scarsini (1989), among other authors. Motivated by this fact, we introduce new stochastic orderings, as well as new social inequality indices to compare the distribution of the wealth in two populations, which could be considered as natural extensions of the first two dominance rules and the S-Gini indices, respectively.

Weak limit for the stationary distribution of the infinite-alleles model

1979 ◽  
Vol 16 (03) ◽  
pp. 459-472
Author(s):  
Haim Avni
Keyword(s):  
Laplace Transform ◽  
Order Statistics ◽  
Frequency Spectrum ◽  
Stationary Distribution ◽  
Weak Limit ◽  
Limit Behavior ◽  
Ewens Sampling Formula ◽  
Sampling Formula ◽  
Spectrum Density ◽  
Infinite Alleles Model

The limit behavior of the stationary distribution of the infinite-alleles model is reduced to a single Laplace transform formula. Some known results, such as Ewens' sampling formula, the distribution of the order-statistics and the frequency spectrum density are shown to follow from this relation. All the results are obtained within the framework of the configuration process, without recourse to finite alleles models. In view of some recent results by Kingman (1977), the results apply wherever the Ewens' sampling formula is valid.

scholarly journals Inverse Stochastic Dominance, Majorization, and Mean Order Statistics

2010 ◽  
Vol 47 (01) ◽  
pp. 277-292 ◽  
Author(s):  
Jesús De La Cal ◽  
Javier Cárcamo
Keyword(s):  
Social Inequality ◽  
Order Statistics ◽  
Stochastic Dominance ◽  
Natural Extension ◽  
Stochastic Order ◽  
Stochastic Orderings ◽  
Natural Extensions ◽  
Weak Majorization ◽  
Two Populations ◽  
Inverse Stochastic Dominance

The inverse stochastic dominance of degree r is a stochastic order of interest in several branches of economics. We discuss it in depth, the central point being the characterization in terms of the weak r-majorization of the vectors of expected order statistics. The weak r-majorization (a notion introduced in the paper) is a natural extension of the classical (reverse) weak majorization of Hardy, Littlewood and Pòlya. This work also shows the equivalence between the continuous majorization (of higher order) and the discrete r-majorization. In particular, our results make it clear that the cases r = 1, 2 differ substantially from those with r ≥ 3, a fact observed earlier by Muliere and Scarsini (1989), among other authors. Motivated by this fact, we introduce new stochastic orderings, as well as new social inequality indices to compare the distribution of the wealth in two populations, which could be considered as natural extensions of the first two dominance rules and the S-Gini indices, respectively.

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