Stochastic orders and majorization of mean order statistics

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.

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Stochastic Comparisons of Symmetric Supermodular Functions of Heterogeneous Random Vectors

Journal of Applied Probability ◽

10.1017/s0021900200013498 ◽

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.

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On a characterization of right spread order by the increasing convex order

Statistics & Probability Letters ◽

10.1016/s0167-7152(99)00048-6 ◽

1999 ◽

Vol 45 (2) ◽

pp. 103-110 ◽

Cited By ~ 30

Author(s):

F. Belzunce

Keyword(s):

Convex Order ◽

Increasing Convex Order ◽

Right Spread Order

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

Journal of Applied Probability ◽

10.1017/jpr.2017.27 ◽

2017 ◽

Vol 54 (3) ◽

pp. 685-700 ◽

Cited By ~ 3

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.

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A characterization of the geometric distribution

Journal of Applied Probability ◽

10.1017/s0021900200097102 ◽

1983 ◽

Vol 20 (01) ◽

pp. 209-212 ◽

Cited By ~ 1

Author(s):

M. Sreehari

Keyword(s):

Positive Integer ◽

Order Statistics ◽

Geometric Distribution ◽

Random Variables ◽

Random Variable ◽

Characterization Problem ◽

Independent Identically Distributed

Let X 1, X 2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X 1:n , X 2:n , …, X n:n . We prove that if the random variable X2:n – X 1:n is independent of the events [X1:n = m] and [X1:n = k], for fixed k > m > 1, then the Xi 's are geometric. This is related to a characterization problem raised by Arnold (1980).

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EQUIVALENT CHARACTERIZATIONS ON ORDERINGS OF ORDER STATISTICS AND SAMPLE RANGES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964809990258 ◽

2010 ◽

Vol 24 (2) ◽

pp. 245-262 ◽

Cited By ~ 33

Author(s):

Tiantian Mao ◽

Taizhong Hu

Keyword(s):

Order Statistics ◽

Exponential Distribution ◽

Geometric Distribution ◽

Random Variables ◽

The Other ◽

Stochastic Orders ◽

Exponential Random Variables ◽

Discrete Counterpart ◽

Similarities And Differences ◽

Geometric Variables

The purpose of this article is to present several equivalent characterizations of comparing the largest-order statistics and sample ranges of two sets of n independent exponential random variables with respect to different stochastic orders, where the random variables in one set are heterogeneous and the random variables in the other set are identically distributed. The main results complement and extend several known results in the literature. The geometric distribution can be regarded as the discrete counterpart of the exponential distribution. We also study the orderings of the largest-order statistics from geometric random variables and point out similarities and differences between orderings of the largest-order statistics from geometric variables and from exponential variables.

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DISPERSION-TYPE VARIABILITY ORDERS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964803173020 ◽

2003 ◽

Vol 17 (3) ◽

pp. 305-334 ◽

Cited By ~ 14

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.

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A characterization of the geometric distribution

Journal of Applied Probability ◽

10.2307/3213738 ◽

1983 ◽

Vol 20 (1) ◽

pp. 209-212 ◽

Cited By ~ 8

Author(s):

M. Sreehari

Keyword(s):

Positive Integer ◽

Order Statistics ◽

Geometric Distribution ◽

Random Variables ◽

Random Variable ◽

Characterization Problem ◽

Independent Identically Distributed

Let X1, X2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X1:n, X2:n, …, Xn:n. We prove that if the random variable X2:n – X1:n is independent of the events [X1:n = m] and [X1:n = k], for fixed k > m > 1, then the Xi's are geometric. This is related to a characterization problem raised by Arnold (1980).

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On the Increasing Convex Order of Relative Spacings of Order Statistics

Mathematics ◽

10.3390/math9060618 ◽

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.

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On a characterization of the exponential distribution by order statistics

Journal of Applied Probability ◽

10.2307/3212540 ◽

1976 ◽

Vol 13 (4) ◽

pp. 818-822 ◽

Cited By ~ 9

Author(s):

M. Ahsanullah

Keyword(s):

Probability Density Function ◽

Probability Density ◽

Order Statistics ◽

Density Function ◽

Exponential Distribution ◽

Random Sample ◽

Random Variables ◽

Identical Distribution

Let X1, X2, …, Xn be a random sample of size n from a population with probability density function f(x), x >0, and let X1,n < X2,n < … < Xn,n be the associated order statistics. A characterization of the exponential distribution is shown by considering the identical distribution of the random variables nX1,n and (n − i + 1)(X1,n −; Xi–1,n) for one i and one n with 2 ≦ i ≦ n.

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