Increasing convex order on generalized aggregation of SAI random variables with applications

2017 ◽  
Vol 54 (3) ◽  
pp. 685-700 ◽  
Author(s):  
Xiaoqing Pan ◽  
Xiaohu Li
Keyword(s):  
Linear Combination ◽  
Kernel Function ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Convex Order ◽  
Increasing Convex Order ◽  
Special Cases ◽  
Arrangement Increasing

Abstract In this paper we study general aggregation of stochastic arrangement increasing random variables, including both the generalized linear combination and the standard aggregation as special cases. In terms of monotonicity, supermodularity, and convexity of the kernel function, we develop several sufficient conditions for the increasing convex order on the generalized aggregations. Some applications in reliability and risks are also presented.

scholarly journals Stochastic Comparisons of Symmetric Supermodular Functions of Heterogeneous Random Vectors

2013 ◽  
Vol 50 (02) ◽  
pp. 464-474
Author(s):  
Antonio Di Crescenzo ◽  
Esther Frostig ◽  
Franco Pellerey
Keyword(s):  
Queueing Theory ◽  
Queueing Networks ◽  
Random Variables ◽  
Random Vectors ◽  
Convex Order ◽  
Stochastic Comparisons ◽  
Increasing Convex Order ◽  
Supermodular Functions ◽  
Stop Loss ◽  
Series Systems

Consider random vectors formed by a finite number of independent groups of independent and identically distributed random variables, where those of the last group are stochastically smaller than those of the other groups. Conditions are given such that certain functions, defined as suitable means of supermodular functions of the random variables of the vectors, are supermodular or increasing directionally convex. Comparisons based on the increasing convex order of supermodular functions of such random vectors are also investigated. Applications of the above results are then provided in risk theory, queueing theory, and reliability theory, with reference to (i) net stop-loss reinsurance premiums of portfolios from different groups of insureds, (ii) closed cyclic multiclass Gordon-Newell queueing networks, and (iii) reliability of series systems formed by units selected from different batches.

scholarly journals Injectivity of linear combinations in $\mathcal{B}(\mathcal{H})$

2021 ◽  
Vol 37 ◽  
pp. 359-369
Author(s):  
Marko Kostadinov
Keyword(s):  
Linear Combination ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Combinations ◽  
Special Cases ◽  
Sufficient And Necessary Conditions ◽  
Alpha Beta ◽  
Necessary And Sufficient

The aim of this paper is to provide sufficient and necessary conditions under which the linear combination $\alpha A + \beta B$, for given operators $A,B \in {\cal B}({\cal H})$ and $\alpha, \beta \in \mathbb{C}\setminus \lbrace 0 \rbrace$, is injective. Using these results, necessary and sufficient conditions for left (right) invertibility are given. Some special cases will be studied as well.

Asymptotic Normality of a Class of Nonparametric Statistics

1987 ◽  
Vol 3 (3) ◽  
pp. 313-347 ◽  
Author(s):  
Munsup Seoh ◽  
Madan L. Puri
Keyword(s):  
Asymptotic Normality ◽  
Order Statistics ◽  
Linear Combination ◽  
Linear Function ◽  
Random Variables ◽  
Weighted Sum ◽  
Rank Statistics ◽  
Concomitants Of Order Statistics ◽  
Signed Rank ◽  
Special Cases

Asymptotic normality is established for a class of statistics which includes as special cases weighted sum of independent and identically distributed (i.i.d.) random variables, unsigned linear rank statistics, signed rank statistics, linear combination of functions of order statistics, and linear function of concomitants of order statistics. The results obtained unify as well as extend a number of known results.

scholarly journals Stochastic orders and majorization of mean order statistics

2006 ◽  
Vol 43 (03) ◽  
pp. 704-712 ◽  
Author(s):  
Jesús de la Cal ◽  
Javier Cárcamo
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Stochastic Orders ◽  
Convex Order ◽  
Increasing Convex Order ◽  
Finite Dimensional ◽  
Integrable Functions

We characterize the (continuous) majorization of integrable functions introduced by Hardy, Littlewood, and Pólya in terms of the (discrete) majorization of finite-dimensional vectors, introduced by the same authors. The most interesting version of this result is the characterization of the (increasing) convex order for integrable random variables in terms of majorization of vectors of expected order statistics. Such a result includes, as particular cases, previous results by Barlow and Proschan and by Alzaid and Proschan, and, in a sense, completes the picture of known results on order statistics. Applications to other stochastic orders are also briefly considered.

STOCHASTIC EQUIVALENCE OF CONVEX ORDERED DISTRIBUTIONS AND APPLICATIONS

2000 ◽  
Vol 14 (1) ◽  
pp. 33-48 ◽  
Author(s):  
M. C. Bhattacharjee ◽  
R. N. Bhattacharya
Keyword(s):  
Recent Result ◽  
Real Line ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Survival Times ◽  
The Real ◽  
Ordered Alternatives ◽  
Special Cases ◽  
Stochastic Equivalence

We consider sufficient conditions for stochastic equivalence of convex ordered random variables. Our main results apply to all convex ordered distributions on the real line and improve on a recent result of Huang and Lin [8] for equality in distribution of convex ordered survival times. Illustrative applications include testing for equality in distribution with convex ordered alternatives and demonstrating several earlier results on stochastic equivalence as special cases.

scholarly journals On the Increasing Convex Order of Relative Spacings of Order Statistics

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 618
Author(s):  
Antonia Castaño-Martínez ◽  
Gema Pigueiras ◽  
Miguel A. Sordo
Keyword(s):  
Order Statistics ◽  
Relative Deprivation ◽  
Sufficient Conditions ◽  
Convex Order ◽  
Increasing Convex Order ◽  
Income Distributions ◽  
Parametric Families ◽  
Relative Differences

Relative spacings are relative differences between order statistics. In this context, we extend previous results concerning the increasing convex order of relative spacings of two distributions from the case of consecutive spacings to general spacings. The sufficient conditions are given in terms of the expected proportional shortfall order. As an application, we compare relative deprivation within some parametric families of income distributions.

scholarly journals Stochastic orders and majorization of mean order statistics

2006 ◽  
Vol 43 (3) ◽  
pp. 704-712 ◽  
Author(s):  
Jesús de la Cal ◽  
Javier Cárcamo
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Stochastic Orders ◽  
Convex Order ◽  
Increasing Convex Order ◽  
Finite Dimensional ◽  
Integrable Functions

We characterize the (continuous) majorization of integrable functions introduced by Hardy, Littlewood, and Pólya in terms of the (discrete) majorization of finite-dimensional vectors, introduced by the same authors. The most interesting version of this result is the characterization of the (increasing) convex order for integrable random variables in terms of majorization of vectors of expected order statistics. Such a result includes, as particular cases, previous results by Barlow and Proschan and by Alzaid and Proschan, and, in a sense, completes the picture of known results on order statistics. Applications to other stochastic orders are also briefly considered.

On sufficient conditions for the comparison in the excess wealth order and spacings

2016 ◽  
Vol 53 (1) ◽  
pp. 33-46 ◽  
Author(s):  
Félix Belzunce ◽  
Carolina Martínez-Riquelme ◽  
José M. Ruiz ◽  
Miguel A. Sordo
Keyword(s):  
Order Statistics ◽  
Sufficient Conditions ◽  
The Other ◽  
Generalized Order Statistics ◽  
Convex Order ◽  
Increasing Convex Order ◽  
Dispersive Order ◽  
Excess Wealth Order ◽  
The One ◽  
Quantities Of Interest

Abstract The purpose of this paper is twofold. On the one hand, we provide sufficient conditions for the excess wealth order. These conditions are based on properties of the quantile functions which are useful when the dispersive order does not hold. On the other hand, we study sufficient conditions for the comparison in the increasing convex order of spacings of generalized order statistics. These results will be combined to show how we can provide comparisons of quantities of interest in reliability and insurance.

scholarly journals Stochastic Comparisons of Symmetric Supermodular Functions of Heterogeneous Random Vectors

2013 ◽  
Vol 50 (2) ◽  
pp. 464-474 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Esther Frostig ◽  
Franco Pellerey
Keyword(s):  
Queueing Theory ◽  
Queueing Networks ◽  
Random Variables ◽  
Random Vectors ◽  
Convex Order ◽  
Stochastic Comparisons ◽  
Increasing Convex Order ◽  
Supermodular Functions ◽  
Stop Loss ◽  
Series Systems

Consider random vectors formed by a finite number of independent groups of independent and identically distributed random variables, where those of the last group are stochastically smaller than those of the other groups. Conditions are given such that certain functions, defined as suitable means of supermodular functions of the random variables of the vectors, are supermodular or increasing directionally convex. Comparisons based on the increasing convex order of supermodular functions of such random vectors are also investigated. Applications of the above results are then provided in risk theory, queueing theory, and reliability theory, with reference to (i) net stop-loss reinsurance premiums of portfolios from different groups of insureds, (ii) closed cyclic multiclass Gordon-Newell queueing networks, and (iii) reliability of series systems formed by units selected from different batches.

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