Journal ArticleDOI

# Stochastic comparisons of weighted sums of arrangement increasing random variables

, Min Yuan1
01 Jul 2015-Statistics & Probability Letters (North-Holland)-Vol. 102, pp 42-50
TL;DR: In this article, the joint density of random variables X 1, X 2, X 3, X 4, X 5, X 6, X n is assumed to be arrangement increasing.

### 1 Introduction

• During the past few decades, linear combinations of random variables have been extensively studied in statistics, operations research, reliability theory, actuarial science and other fields.
• They proved that characterizations of these notions are related to properties of arrangement increasing (AI) functions (to be defined in Section 2).

### 2 Preliminaries

• The authors give definitions of some stochastic orders, majorization orders and supermodular [submodular] functions.
• Let X and Y be two random variables with probability (mass) density functions f and g; and survival functions F and G respectively.

### 3 Main results

• The authors study stochastic comparisons of weighted sums of the form ∑n i=1 φ(Xi, ai) where X1, . . . , Xn are random variables with joint density function f(x).
• The authors consider both usual stochastic order as well as increasing convex order for comparison purposes.

### 3.1 Usual stochastic ordering

• Before the authors give the main result, they list several lemmas, which will be used in the sequel.
• Since f(x) is log-concave, f(x)IA is log-concave.

### 3.2 Increasing convex ordering

• So far, the authors have obtained all the results under the assumption (A1) that the joint density function is log-concave.
• Therefore, (3.6) holds and the desired result follows.

### 4 An application to optimal capital allocation

• The authors outline an application of their main results.
• Therefore, it is meaningful for us to find the best capital allocation strategy if it exists via the methods in Section 3.
• Without loss of generality, the authors assume p1 ≤ p2.

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### Cites background from "Stochastic comparisons of weighted ..."

• ...The following characterization was pointed out in Shanthikumar andYao (1991) and further remarked upon in Cai and Wei (2014) and Pan et al. (2015)....

[...]

##### References
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Book
06 Apr 2011
TL;DR: In this paper, Doubly Stochastic Matrices and Schur-Convex Functions are used to represent matrix functions in the context of matrix factorizations, compounds, direct products and M-matrices.
Abstract: Introduction.- Doubly Stochastic Matrices.- Schur-Convex Functions.- Equivalent Conditions for Majorization.- Preservation and Generation of Majorization.- Rearrangements and Majorization.- Combinatorial Analysis.- Geometric Inequalities.- Matrix Theory.- Numerical Analysis.- Stochastic Majorizations.- Probabilistic, Statistical, and Other Applications.- Additional Statistical Applications.- Orderings Extending Majorization.- Multivariate Majorization.- Convex Functions and Some Classical Inequalities.- Stochastic Ordering.- Total Positivity.- Matrix Factorizations, Compounds, Direct Products, and M-Matrices.- Extremal Representations of Matrix Functions.

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### "Stochastic comparisons of weighted ..." refers background in this paper

• ...For extensive and comprehensive details on the theory of majorization order and their applications, please refer to Marshall et al. (2011)....

[...]

• ...3 Hollander et al. (1977) and Marshall et al. (2011) gave many examples of AI random variables....

[...]

• ...If ϕ : <n → < has second partial derivatives, then it is supermodular [submodular] if and 4 only if ∂2 ∂xi∂xj ϕ(x) ≥ [≤] 0 for all i 6= j and x ∈ <n. Marshall et al. (2011) gave several examples of supermodelar [submodular] functions....

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TL;DR: In this article, the authors present an univariate Stochastic model for queuing systems and compare its properties with those of other non-stochastic models and compare risks.
Abstract: Preface. Univariate Stochastic Orders Theory of Integral Stochastic Orders Multivariate Stochastic Orders Stochastic Models, Comparison and Monotonicity Monotonicity and Comparability of Stochastic Processes Monotonicity Properties and Bounds for Queueing Systems Applications to Various Stochastic Models Comparing Risks. List of Symbols. References. Index.

1,739 citations

Journal ArticleDOI
TL;DR: The author does an admirable job of explaining the differences between Bayesian probability and the frequentist notion of probability, showing that, philosophically, only the Bayesian makes sense.
Abstract: (2003). Comparison Methods for Stochastic Models and Risks. Technometrics: Vol. 45, No. 4, pp. 370-371.

611 citations

### "Stochastic comparisons of weighted ..." refers background in this paper

• ...The relationships among these orders are shown in the following diagram (see Shaked and Shanthikumar, 2007; Müller and Stoyan, 2002): X ≤lr Y =⇒ X ≤hr Y =⇒ X ≤st Y =⇒ X ≤icx Y. Shanthikumar and Yao (1991) considered the problem of extending the above concepts to compare the components of dependent…...

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TL;DR: In this article, the authors provide a complete characterization of logconcavity, an increasingly popular concept in the economics of uncertainty and information, without assuming that density functions are differentiable.

267 citations

### "Stochastic comparisons of weighted ..." refers background in this paper

• ...We say that X is smaller than Y (1) in the usual stochastic order, denoted by X ≤st Y , if F (t) ≤ G(t) for all t or, equivalently, if E[h(X)] ≤ E[h(Y )] for all increasing functions h; (2) in the hazard rate order, denoted by X ≤hr Y , if G(t)/F (t) is increasing in t for which the ratio is well defined; (3) in the likelihood ratio order, denoted by X ≤lr Y , if g(t)/f(t) is increasing in t for which the ratio is well defined; (4) in the increasing convex order, denoted by X ≤icx Y , if E[h(X)] ≤ E[h(Y )] for all increasing convex functions h for which the expectations exist....

[...]

• ...An (1998) remarked that if X has a log-concave density, then its density has at most an exponential tail, i.e., f(x) = O (exp(−λx)) , λ > 0, x→∞....

[...]

• ...i=3 φ(Xi, ci) ≥st φ(X1, b(1)) + φ(X2, b(2)) + n ∑...

[...]

• ...1) then (x1, x2) m (y1, y2) =⇒ g(x(1), x(2)) ≥ g(y(1), y(2))....

[...]

• ...g(x(2), x(1)) ≥ g(x(1), x(2)) for all (x1, x2) ∈ <(2), (3....

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Journal ArticleDOI
TL;DR: In this paper, the authors provide a general approach that supports the pairwise interchange arguments widely used in various settings, and develop new notions of stochastic order relations so that dependent random variables can be meaningfully compared.
Abstract: Bivariate (and multivariate) functional representations for the following stochastic order relations are established: the likelihood ratio ordering, hazard rate ordering, and the usual stochastic ordering. The motivation of the study is (i) to provide a general approach that supports the 'pairwise interchange' arguments widely used in various settings, and (ii) to develop new notions of stochastic order relations so that dependent random variables can be meaningfully compared. Applications are illustrated through problems in stochastic scheduling, closed

127 citations

### "Stochastic comparisons of weighted ..." refers background in this paper

• ...…among these orders are shown in the following diagram (see Shaked and Shanthikumar, 2007; Müller and Stoyan, 2002): X ≤lr Y =⇒ X ≤hr Y =⇒ X ≤st Y =⇒ X ≤icx Y. Shanthikumar and Yao (1991) considered the problem of extending the above concepts to compare the components of dependent random variables....

[...]

• ...As pointed out by Shanthikumar and Yao (1991), joint likelihood ratio ordering between the components of a bivariate random vector may not imply likelihood ratio ordering between their marginal distributions unless the random variables are independent, but it does imply stochastic ordering between…...

[...]

###### Q1. What have the authors contributed in "Stochastic comparisons of weighted sums of arrangement increasing random variables" ?

In this paper, the joint density of random variables X1, X2, ..., Xn is assumed to be an arrangement increasing ( AI ) function.

During the past few decades, linear combinations of random variables have been extensively studied in statistics, operations research, reliability theory, actuarial science and other fields.

Since the usual stochastic order is closed under convolution, the authors only need to proveφ(X1 − p2) + φ(X2 − p1) ≥st φ(X1 − p1) + φ(X2 − p2). (4.1)Under assumptions (A1) and (A2), (4.1) holds due to Corollary 3.8.

The first author is supported by the NNSF of China (No. 11401558), the Fundamental Research Funds for the Central Universities (No. WK2040160010) and China Postdoctoral12Science Foundation (No. 2014M561823).

If the joint density function of X1, . . . , Xn satisfies assumptions (A1) and (A2) of Section 3, and if p∗ = (p∗1, . . . , p ∗ n) is the solution to the best capital allocation strategy, then, the authors have p∗1 ≤ p∗2 ≤ . . . ≤ p∗n .

As defined in Xu and Hu (2012), the loss functionL(p) = n∑ i=1 φ(Xi − pi), p ∈ A = {p ∈ <n+ : p1 + . . .+ pn = p}is a reasonable criterion to set the capital amount pi to Xi, where φ is convex.

An (1998) remarked that if X has a log-concave density, then its density has at most an exponential tail, i.e.,f(x) = O (exp(−λx)) , λ > 0, x→∞.9

In this section, the authors study stochastic comparisons of weighted sums of the form ∑ni=1 φ(Xi, ai)where X1, . . . , Xn are random variables with joint density function f(x).

Then g is log-concave on <m.5Lemma 3.2 (Pan et al., 2013)If g : <2 → <+ is log-concave and −g is AI, i.e.g(x(2), x(1)) ≥ g(x(1), x(2)) for all (x1, x2) ∈ <2, (3.1)then(x1, x2) m (y1, y2) =⇒ g(x(1), x(2)) ≥ g(y(1), y(2)).

They proved that characterizations of these notions are related to properties of arrangement increasing (AI) functions (to be defined in Section 2).

In this paper the authors further study the problem of stochastic comparisons of linear combinations of AI random variables not only for increasing convex ordering, but also for the usual stochastic ordering.

it is sufficient to proveh (ϕ (x,a12)) + h (ϕ (x,a21))− h (ϕ (x,b12))− h (ϕ (x,b21)) ≥ 0. (3.6)Since φ is convex and supermodular, for a(1) ≤ b(1) ≤ b(2) ≤ a(2) and x1 ≤ x2, the authors haveϕ(x,a12)− ϕ(x,b12) = φ(x1, a(1)) + φ(x2, a(2))− φ(x1, b(1))− φ(x2, b(2)) = [ φ(x1, a(1)) + φ(x1, a(2))− φ(x1, b(1))− φ(x1, b(2)) ] + [ φ(x2, a(2))− φ(x1, a(2)) + φ(x1, b(2))− φ(x2, b(2)) ] ≥ 0,ϕ(x,a12)− ϕ(x,b21) = φ(x1, a(1)) + φ(x2, a(2))− φ(x2, b(1))− φ(x1, b(2)) = [ φ(x1, a(1)) + φ(x1, a(2))− φ(x1, b(1))− φ(x1, b(2)) ] + [ φ(x2, a(2))− φ(x1, a(2)) + φ(x1, b(1))− φ(x2, b(1)) ] ≥ 0andϕ(x,a12) + ϕ(x,a21)− ϕ(x,b12)− ϕ(x,b21)= φ(x1, a(1)) + φ(x1, a(2))− φ(x1, b(1))− φ(x1, b(2))+φ(x2, a(2)) + φ(x2, a(1))− φ(x2, b(2))− φ(x2, b(1)) ≥ 0.Thus, for any increasing convex function h, if ϕ(x,a21) ≥ ϕ(x,b21), then h(ϕ(x,a21)) ≥ h(ϕ(x,b21)) and h(ϕ(x,a12)) ≥ h(ϕ(x,b12)), which implies (3.6).