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.

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 dominance relations for generalised parametric distributions obtained through composition

METRON ◽  
2020 ◽  
Vol 78 (3) ◽  
pp. 297-311
Author(s):  
Tommaso Lando ◽  
Lucio Bertoli-Barsotti
Keyword(s):  
Order Statistics ◽  
Stochastic Dominance ◽  
Convex Order ◽  
Increasing Convex Order ◽  
Dominance Relations ◽  
Order Stochastic Dominance ◽  
Functional Forms ◽  
Parametric Families ◽  
Second Order Stochastic Dominance ◽  
Generated Family

AbstractInvestigating stochastic dominance within flexible multi-parametric families of distributions is often complicated, owing to the high number of parameters or non-closed functional forms. To simplify the problem, we use the T–X method, making it possible to obtain generalised models through the composition of cumulative distributions and quantile functions. We derive conditions for the second-order stochastic dominance and for the increasing convex order within multi-parametric families in two steps, namely: (i) breaking them down via the T–X approach and (ii) checking dominance conditions of the (more) manageable distributions composing the model. We apply our method to some special distributions and focus on the beta-generated family, which enables the comparisons of order statistics of i.i.d. samples from (possibly) different random variables.

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 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.

scholarly journals Multivariate Classes of GB2 Distributions with Applications

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 72
Author(s):  
José María Sarabia ◽  
Vanesa Jordá ◽  
Faustino Prieto ◽  
Montserrat Guillén
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Multivariate Distributions ◽  
Empirical Application ◽  
Income Distributions ◽  
Flexible Distribution ◽  
Parametric Families ◽  
Wind Events

The general beta of the second kind distribution (GB2) is a flexible distribution which includes several relevant parametric families of distributions. This distribution has important applications in earnings and income distributions, finance and insurance. In this paper, several multivariate classes of the GB2 distribution are proposed. The different multivariate versions are based on two simple univariate representations of the GB2 distribution. The first type of multivariate distributions are constructed from a stochastic dependent representations defined in terms of gamma random variables. Using this representation and beginning by two particular multivariate GB2 distributions, multivariate Singh–Maddala and Dagum income distributions are presented and several properties are obtained. Then, a general multivariate GB2 distribution is introduced. The second type of multivariate distributions are based on a generalization of the distribution of the order statistics, which gives place to multivariate GB2 distribution with support above the diagonal. We discuss the role of these families in modeling bivariate income distributions. Finally, an empirical application is given, where we show that a multivariate GB2 distribution can be useful for modeling compound precipitation and wind events in the whole range.

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.

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.

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.

Order statistics, Poisson processes and repairable systems

1976 ◽  
Vol 13 (03) ◽  
pp. 519-529 ◽  
Author(s):  
Douglas R. Miller
Keyword(s):  
Order Statistics ◽  
Point Processes ◽  
Failure Time ◽  
Sufficient Conditions ◽  
Renewal Processes ◽  
Repairable Systems ◽  
Homogeneous Poisson Process ◽  
Necessary And Sufficient ◽  
Non Homogeneous Poisson Process ◽  
Weibull Process

Necessary and sufficient conditions are presented under which the point processes equivalent to order statistics of n i.i.d. random variables or superpositions of n i.i.d. renewal processes converge to a non-degenerate limiting process as n approaches infinity. The limiting process must be one of three types of non-homogeneous Poisson process, one of which is the Weibull process. These point processes occur as failure-time models in the reliability theory of repairable systems.

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