scholarly journals On the Dynamic Cumulative Past Quantile Entropy Ordering

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2001
Author(s):  
Haiyan Wang ◽  
Diantong Kang ◽  
Lei Yan
Keyword(s):  
Stochastic Models ◽  
Natural Science ◽  
Stochastic Order ◽  
Random Variables ◽  
Uncertainty Management ◽  
Stochastic Orders ◽  
New Order ◽  
Closure Properties ◽  
Life Distributions

In many society and natural science fields, some stochastic orders have been established in the literature to compare the variability of two random variables. For a stochastic order, if an individual (or a unit) has some property, sometimes we need to infer that the population (or a system) also has the same property. Then, we say this order has closed property. Reversely, we say this order has reversed closure. This kind of symmetry or anti-symmetry is constructive to uncertainty management. In this paper, we obtain a quantile version of DCPE, termed as the dynamic cumulative past quantile entropy (DCPQE). On the basis of the DCPQE function, we introduce two new nonparametric classes of life distributions and a new stochastic order, the dynamic cumulative past quantile entropy (DCPQE) order. Some characterization results of the new order are investigated, some closure and reversed closure properties of the DCPQE order are obtained. As applications of one of the main results, we also deal with the preservation of the DCPQE order in several stochastic models.

scholarly journals Further Results of the TTT Transform Ordering of Order n

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1960
Author(s):  
Lei Yan ◽  
Diantong Kang ◽  
Haiyan Wang
Keyword(s):  
Reliability Analysis ◽  
Partial Order ◽  
Stochastic Models ◽  
Stochastic Order ◽  
Order Relation ◽  
Partial Order Relation ◽  
Coherent Systems ◽  
Closure Properties ◽  
Distributed Components ◽  
Distribution Class

To compare the variability of two random variables, we can use a partial order relation defined on a distribution class, which contains the anti-symmetry. Recently, Nair et al. studied the properties of total time on test (TTT) transforms of order n and examined their applications in reliability analysis. Based on the TTT transform functions of order n, they proposed a new stochastic order, the TTT transform ordering of order n (TTT-n), and discussed the implications of order TTT-n. The aim of the present study is to consider the closure and reversed closure of the TTT-n ordering. We examine some characterizations of the TTT-n ordering, and obtain the closure and reversed closure properties of this new stochastic order under several reliability operations. Preservation results of this order in several stochastic models are investigated. The closure and reversed closure properties of the TTT-n ordering for coherent systems with dependent and identically distributed components are also obtained.

The total time on test transform and the excess wealth stochastic orders of distributions

2002 ◽  
Vol 34 (04) ◽  
pp. 826-845 ◽  
Author(s):  
Subhash C. Kochar ◽  
Xiaohu Li ◽  
Moshe Shaked
Keyword(s):  
Statistical Theory ◽  
Stochastic Order ◽  
Random Variables ◽  
Distribution Functions ◽  
Stochastic Orders ◽  
Right Spread Order

For nonnegative random variables X and Y we write X ≤TTT Y if ∫0 F -1(p)(1-F(x))dx ≤ ∫0 G -1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.

Some generalized variability orderings among life distributions with reliability applications

1991 ◽  
Vol 28 (02) ◽  
pp. 374-383 ◽  
Author(s):  
M. C. Bhattacharjee
Keyword(s):  
Random Variables ◽  
Parallel Systems ◽  
Reliability Theory ◽  
Dominance Relation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closure Properties ◽  
Life Distributions ◽  
The Mean ◽  
Necessary And Sufficient

We investigate a generalized variability ordering and its weaker versions among non-negative random variables (lifetimes of components). Our results include a necessary and sufficient condition which justifies the generalized variability interpretation of this dominance relation between life distributions, relationships to some weakly aging classes in reliability theory, closure properties and inequalities for the mean life of series and parallel systems under such ordering.

scholarly journals Reliability Analysis of the Proportional Mean Residual Life Order

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
M. Kayid ◽  
S. Izadkhah ◽  
H. Alhalees
Keyword(s):  
Residual Life ◽  
Stochastic Order ◽  
Mean Residual Life ◽  
New Order ◽  
Mean Residual Life Function ◽  
New Class ◽  
Mean Residual Life Order ◽  
Life Distributions ◽  
The Mean ◽  
Life Function

The concept of mean residual life plays an important role in reliability and life testing. In this paper, we introduce and study a new stochastic order called proportional mean residual life order. Several characterizations and preservation properties of the new order under some reliability operations are discussed. As a consequence, a new class of life distributions is introduced on the basis of the anti-star-shaped property of the mean residual life function. We study some reliability properties and some characterizations of this class and provide some examples of interest in reliability.

DISPERSION-TYPE VARIABILITY ORDERS

2003 ◽  
Vol 17 (3) ◽  
pp. 305-334 ◽  
Author(s):  
Félix Belzunce ◽  
Taizhong Hu ◽  
Baha-Eldin Khaledi
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Stochastic Orders ◽  
Closure Properties ◽  
Dispersive Order ◽  
Right Spread Order ◽  
The Right

Dispersion-type orders are introduced and studied. The new orders can be used to compare the variability of the underlying random variables, among which are the usual dispersive order and the right spread order. Connections among the new orders and other common stochastic orders are examined and investigated. Some closure properties of the new orders under the operation of order statistics, transformations, and mixtures are derived. Finally, several applications of the new orders are given.

scholarly journals On the Generalized Decreasing Mean Time to Failure or Replaced Ordering

2021 ◽  
Vol 2021 ◽  
pp. 1-23
Author(s):  
Haiyan Wang ◽  
Diantong Kang ◽  
Lei Yan
Keyword(s):  
Stochastic Orders ◽  
New Order ◽  
Time To Failure ◽  
Closure Properties ◽  
Mean Time To Failure ◽  
Mean Time

In this paper, we establish two new stochastic orders, DMTFR (decreasing mean time to failure or replaced) and GDMTFR (generalized decreasing mean time to failure or replaced), and mainly investigate properties of the GDMTFR order. Some characterizations of the GDMTFR order are given. The implication relationships between the DMTFR and the GDMTFR orders are considered. Also, closure and reversed closure properties of the new order GDMTFR are investigated. Meanwhile, several illustrative examples that meet the GDMTFR order are shown as well.

The total time on test transform and the excess wealth stochastic orders of distributions

2002 ◽  
Vol 34 (4) ◽  
pp. 826-845 ◽  
Author(s):  
Subhash C. Kochar ◽  
Xiaohu Li ◽  
Moshe Shaked
Keyword(s):  
Statistical Theory ◽  
Stochastic Order ◽  
Random Variables ◽  
Distribution Functions ◽  
Stochastic Orders ◽  
Right Spread Order

For nonnegative random variables X and Y we write X ≤TTTY if ∫0F-1(p)(1-F(x))dx ≤ ∫0G-1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.

RESIDUAL STOCHASTIC PRECEDENCE ORDER

2020 ◽  
pp. 1-17
Author(s):  
Amit Kumar Misra ◽  
Vaishali Gupta ◽  
Neeraj Misra
Keyword(s):  
Stochastic Order ◽  
Random Variables ◽  
Reliability Theory ◽  
Stochastic Orders ◽  
Dependent Random Variables ◽  
Stochastic Precedence ◽  
Residual Lifetimes

The aim of this paper is to introduce a new stochastic order based on the residual lifetimes of two nonnegative dependent random variables and the stochastic precedence order. We develop some characterizations and preservation properties of this stochastic order. In addition, we study some of its reliability properties and its relation with other existing stochastic orders. One of the possible applications in reliability theory has also been discussed.

Some generalized variability orderings among life distributions with reliability applications

1991 ◽  
Vol 28 (2) ◽  
pp. 374-383 ◽  
Author(s):  
M. C. Bhattacharjee
Keyword(s):  
Random Variables ◽  
Parallel Systems ◽  
Reliability Theory ◽  
Dominance Relation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closure Properties ◽  
Life Distributions ◽  
The Mean ◽  
Necessary And Sufficient

We investigate a generalized variability ordering and its weaker versions among non-negative random variables (lifetimes of components). Our results include a necessary and sufficient condition which justifies the generalized variability interpretation of this dominance relation between life distributions, relationships to some weakly aging classes in reliability theory, closure properties and inequalities for the mean life of series and parallel systems under such ordering.

scholarly journals Stochastic Order and Generalized Weighted Mean Invariance

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 662
Author(s):  
Mateu Sbert ◽  
Jordi Poch ◽  
Shuning Chen ◽  
Víctor Elvira
Keyword(s):  
Monte Carlo ◽  
Probability Distributions ◽  
Stochastic Order ◽  
Arithmetic Mean ◽  
Stochastic Orders ◽  
Arithmetic Means ◽  
Increasing Convex Order ◽  
Monte Carlo Techniques ◽  
Multiple Importance Sampling ◽  
Invariance Properties

In this paper, we present order invariance theoretical results for weighted quasi-arithmetic means of a monotonic series of numbers. The quasi-arithmetic mean, or Kolmogorov–Nagumo mean, generalizes the classical mean and appears in many disciplines, from information theory to physics, from economics to traffic flow. Stochastic orders are defined on weights (or equivalently, discrete probability distributions). They were introduced to study risk in economics and decision theory, and recently have found utility in Monte Carlo techniques and in image processing. We show in this paper that, if two distributions of weights are ordered under first stochastic order, then for any monotonic series of numbers their weighted quasi-arithmetic means share the same order. This means for instance that arithmetic and harmonic mean for two different distributions of weights always have to be aligned if the weights are stochastically ordered, this is, either both means increase or both decrease. We explore the invariance properties when convex (concave) functions define both the quasi-arithmetic mean and the series of numbers, we show its relationship with increasing concave order and increasing convex order, and we observe the important role played by a new defined mirror property of stochastic orders. We also give some applications to entropy and cross-entropy and present an example of multiple importance sampling Monte Carlo technique that illustrates the usefulness and transversality of our approach. Invariance theorems are useful when a system is represented by a set of quasi-arithmetic means and we want to change the distribution of weights so that all means evolve in the same direction.

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