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.

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.

Univariate and multivariate stochastic orderings of residual lifetimes of live components in sequential (𝑛-𝑟+ 1)-out-of-𝑛 systems

2018 ◽  
Vol 55 (3) ◽  
pp. 834-844
Author(s):  
Ghobad Barmalzan ◽  
Abedin Haidari ◽  
Narayanaswamy Balakrishnan
Keyword(s):  
Order Statistics ◽  
Exponential Distribution ◽  
Sufficient Conditions ◽  
Sequential Order ◽  
Stochastic Orderings ◽  
Sequential Order Statistics ◽  
Residual Lifetimes

Abstract Sequential order statistics can be used to describe the ordered lifetimes of components of a system when the failure of a component may affect the reliability of the remaining components. After a reliability system consisting of n components fails, some of its components may still be alive. In this paper we first establish some univariate stochastic orderings and ageing properties of the residual lifetimes of the live components in a sequential (n-r+1)-out-of-n system. We also obtain a characterizing result for the exponential distribution based on uncorrelated residual lifetimes of live components. Finally, we provide some sufficient conditions for comparing vectors of residual lifetimes of the live components from two sequential (n-r+1)-out-of-n systems. The results established here extend some well-known results in the literature.

Many-valued logics of extended Gentzen style II

1970 ◽  
Vol 35 (4) ◽  
pp. 493-528 ◽  
Author(s):  
Moto-o Takahashi
Keyword(s):  
Topological Space ◽  
Model Theory ◽  
Natural Extension ◽  
Preceding Paper ◽  
Considerable Extent ◽  
Continuous Logic ◽  
First Order ◽  
Natural Extensions ◽  
Fundamental Theorems ◽  
Order Continuous

In the monograph [1] of Chang and Keisler, a considerable extent of model theory of the first order continuous logic (that is, roughly speaking, many-valued logic with truth values from a topological space) is ingeniously developed without using any notion of provability.In this paper we shall define the notion of provability in continuous logic as well as the notion of matrix, which is a natural extension of one in finite-valued logic in [2], and develop the syntax and semantics of it mostly along the line in the preceding paper [2]. Fundamental theorems of model theory in continuous logic, which have been proved with purely model-theoretic proofs (i.e. those proofs which do not use any proof-theoretic notions) in [1], will be proved with proofs which are natural extensions of those in two-valued logic.

scholarly journals Stochastic orderings induced by star-shaped functions

1991 ◽  
Vol 14 (4) ◽  
pp. 639-664
Author(s):  
Henry A. Krieger
Keyword(s):  
Real Line ◽  
Stochastic Dominance ◽  
Joint Distribution ◽  
Decomposition Theorem ◽  
Open Interval ◽  
Distribution Functions ◽  
Proper Subset ◽  
Stochastic Orderings ◽  
The Real ◽  
Decreasing Functions

The non-decreasing functions whicl are star-shaped and supported above at each point of a non-empty closed proper subset of the real line induce an ordering, on the class of distribution functions with finite first moments, that is strictly weaker than first degree stochastic dominance and strictly stronger than second degree stochastic dominance. Several characterizations of this ordering are developed, both joint distribution criteria and those involving only marginals. Tle latter are deduced from a decomposition theorem, which reduces the problem to consideration of certain functions which are star-shaped on the complement of an open interval.

scholarly journals Order-Invariant Measures on Fixed Causal Sets

2012 ◽  
Vol 21 (3) ◽  
pp. 330-357 ◽  
Author(s):  
GRAHAM BRIGHTWELL ◽  
MALWINA LUCZAK
Keyword(s):  
Linear Extension ◽  
Invariant Measures ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Order Type ◽  
Probability Measures ◽  
Causal Sets ◽  
Natural Extensions ◽  
Countably Infinite ◽  
Order Invariant

A causal set is a countably infinite poset in which every element is above finitely many others; causal sets are exactly the posets that have a linear extension with the order-type of the natural numbers; we call such a linear extension a natural extension. We study probability measures on the set of natural extensions of a causal set, especially those measures having the property of order-invariance: if we condition on the set of the bottom k elements of the natural extension, each feasible ordering among these k elements is equally likely. We give sufficient conditions for the existence and uniqueness of an order-invariant measure on the set of natural extensions of a causal set.

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.

Sample Path Criteria for Weak Majorization

1994 ◽  
Vol 26 (01) ◽  
pp. 155-171 ◽  
Author(s):  
Panayotis D. Sparaggis ◽  
Don Towsley ◽  
Christos G. Cassandras
Keyword(s):  
Performance Measures ◽  
Queueing Systems ◽  
Sample Path ◽  
Cumulative Number ◽  
Stochastic Orderings ◽  
Optimal Policies ◽  
Weak Majorization ◽  
Queue Lengths ◽  
And Performance

We present two forms of weak majorization, namely, very weak majorization and p-weak majorization that can be used as sample path criteria in the analysis of queueing systems. We demonstrate how these two criteria can be used in making comparisons among the joint queue lengths of queueing systems with blocking and/or multiple classes, by capturing an interesting interaction between state and performance descriptors. As a result, stochastic orderings on performance measures such as the cumulative number of losses can be derived. We describe applications that involve the determination of optimal policies in the context of load-balancing and scheduling.

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