- Email: [email protected]
A new proof for the peakedness of linear combinations of random variables
Statistics and Probability Letters 114 (2016) 93–98
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
A new proof for the peakedness of linear combinations of random variables Shan Ju, Xiaoqing Pan ∗ Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
article
info
Article history: Received 8 October 2015 Received in revised form 14 March 2016 Accepted 14 March 2016 Available online 30 March 2016
abstract A new proof is given to compare linear combinations of independent and possibly non-identically distributed random variables, as well as bivariate SAI random variables in the sense of peakedness order. The main results extend those given in Proschan (1965) and Ma (1998). © 2016 Elsevier B.V. All rights reserved.
MSC: 60E15 62N05 62G30 Keywords: Peakedness Majorization Stochastically arrangement increasing Log-concavity Schur-concave [Schur-convex]
1. Introduction Let X1 , . . . , Xn be independent and identically distributed (i.i.d.) random variables with a symmetric log-concave density function. Theorem 2.3 of Proschan (1965) shows that, for any a = (a1 , . . . , an ), b = (b1 , . . . , bn ) ∈ ℜn+ , it holds
n n a ≽m b =⇒ P aX≥t ≥P bX≥t , i=1 i i i =1 i i
t ≥ 0,
where ‘≽m ’ denotes the majorization order (see Section 2 for the formal definition). This topic has been followed by many researchers because of its wide applications in statistics, reliability theory, economic theory and other fields. For example, Tong (1994) studied various applications of this result in probability and statistics. Ma (1998) generalized this result from i.i.d. random variables to independent and possibly non-identically distributed (i.ni.d.) random variables with symmetric log-concave density functions. The application of this result in economic theory was discussed in Ibragimov (2007). Zhao et al. (2011) and Xu and Hu (2011) studied various sufficient conditions for comparing the distributions of linear combinations of independent random variables. One may also refer to Pan et al. (2013, 2015) and references therein for the recent development of this topic.
∗
Corresponding author. E-mail addresses: [email protected] (S. Ju), [email protected] (X. Pan).
http://dx.doi.org/10.1016/j.spl.2016.03.012 0167-7152/© 2016 Elsevier B.V. All rights reserved.
94
S. Ju, X. Pan / Statistics and Probability Letters 114 (2016) 93–98
It is worth mentioning that all the methodologies in the literature for comparing the distributions of linear combinations of random variables are adopted from that in Proschan (1965). However, the methodology of Proschan (1965) cannot be applied to the case of dependent random variables. In this note, we propose a new proof for Theorem 2.3 of Proschan (1965). The new approach can be used not only for the case of i.ni.d. random variables but also for the case of dependent variables. The rest of the paper is organized as follows. Section 2 recalls the definitions of some stochastic orders, majorization, and stochastic arrangement increasing random vectors. The main results are presented in Section 3. Throughout this paper, the terms increasing and decreasing stand for non-decreasing and non-increasing, respectively, and all expectations are implicitly assumed to be finite whenever they appear. 2. Preliminaries In this section, we recall some notions which are pertinent to our main discussion. Definition 2.1 (Shaked and Shanthikumar, 2007). Let X and Y be two random variables. X is said to be smaller than Y (i) in the usual stochastic order, denoted by X ≤st Y , if P(X > t ) ≤ P(Y > t ) for all t; (ii) 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, where f and g are the density functions of X and Y , respectively. The above stochastic orders can be used to compare the magnitudes of random variables. The relationship between them is well-known in the literature (see Shaked and Shanthikumar, 2007 and Müller and Stoyan, 2002): X ≤lr Y =⇒ X ≤st Y . Definition 2.2 (Birnbaum, 1948). Let X and Y be random variables and let u and v be real constants. X is said to be more peaked about u than Y about v , denoted by X ≤peak Y , if P (|X − u| ≥ t ) ≤ P (|Y − v| ≥ t ) ,
t ≥ 0.
For u = v = 0, we simply say that X is more peaked than Y , or, equivalently, |X | ≤st |Y |. The majorization deals with a partial ordering of the diversity of the components of a vector. For more details on the theory of majorization, see Marshall et al. (2011). Let x(1) ≤ x(2) ≤ · · · ≤ x(n) be the increasing arrangement of the components of the vector x = (x1 , . . . , xn ). Definition 2.3. For vectors x, y ∈ ℜn , x is said to be majorized by y, denoted by x ≼m y, if j i=1
x(i) ≥
j
y(i)
n
i=1
x(i) =
n
i =1
y(i) and
for j = 1, . . . , n − 1.
i=1
A real-valued function φ defined on a set A ⊆ ℜn is said to be Schur-concave [Schur-convex] on A if, for any x, y ∈ A, x ≽m y =⇒ φ(x) ≤ [≥] φ(y ); and φ is said to be log-concave on A ⊂ ℜn if A is a convex set and, for any x, y ∈ A and α ∈ [0, 1],
φ(α x + (1 − α)y ) ≥ [φ(x)]α [φ(y )]1−α . Denote π = (π (1), . . . , π (n)) a permutation of {1, . . . , n}, and define π (x) = (xπ (1) , . . . , xπ (n) ) for x ∈ ℜn . For any 1 ≤ i ̸= j ≤ n, let (πij (1), . . . , πij (n)) with πij (i) = j, πij (j) = i and πij (k) = k for k ̸∈ {i, j}. Definition 2.4. A real-valued function g (x) defined on ℜn is said to be an arrangement increasing (AI) function if
(xi − xj ) g (x) − g (πij (x)) ≤ 0, for all x ∈ ℜn and any pair (i, j) such that 1 ≤ i < j ≤ n. For more details on AI functions and their applications, one may refer to Hollander et al. (1977), and Boland and Proschan (1988). Recently, Cai and Wei (2014, 2015) introduced the following notion to characterize the monotonicity of mutually dependent random variables. Definition 2.5. A random vector X = (X1 , . . . , Xn ) is said to be stochastic arrangement increasing (SAI) if E[g (X )] ≥ E[g (πij (X ))], for any AI function g and all pair (i, j) such that 1 ≤ i < j ≤ n.
S. Ju, X. Pan / Statistics and Probability Letters 114 (2016) 93–98
95
An absolutely continuous SAI random vector can be characterized by its joint probability density as follows. Lemma 2.6. An absolutely continuous random vector is SAI if and only if the corresponding probability density function is AI. This characterization was pointed out in Shanthikumar and Yao (1991) and further discussed in Cai and Wei (2014), and Pan et al. (2015). The examples of SAI random vector include: (1) X1 , . . . , Xn are i.i.d. random variables; (2) X1 , . . . , Xn are exchangeable random variables; (3) X1 , . . . , Xn are independent random variables satisfying X1 ≤lr X2 ≤lr · · · ≤lr Xn . Many well-known distributions are SAI (Hollander et al., 1977), which include multinomial, negative multinomial, multivariate hypergeometric, dirichlet, inverted dirichlet, negative multivariate hypergeometric, dirichlet compound negative multinomial, multivariate logarithmic series distribution, multivariate F distribution, multivariate Pareto distribution, and multivariate normal distribution. 3. Main results In this section, we present a new proof for Theorem 2.3 of Proschan (1965), and also discuss the case of bivariate SAI random variables. The following lemmas are needed to present our main results. Lemma 3.1 (Birnbaum, 1948 and Shaked and Shanthikumar, 2007). Let X1 , . . . , Xn and Y1 , . . . , Yn be two sets of independent random variables with Xi ’s and Yi ’s (i = 1, . . . , n) having distribution functions that are symmetric about possibly different centers and unimodal densities with possibly some probability mass at their respective centers. If Xi ≤peak Yi for i = 1, . . . , n, then n
Xi ≤peak
i =1
n
Yi .
i=1
Lemma 3.2 (Pan et al., 2013). If g : ℜ2 → ℜ+ is log-concave and −g is AI, i.e. g x(2) , x(1) ≥ g x(1) , x(2)
forall (x1 , x2 ) ∈ ℜ2 ,
(3.1)
then
(x1 , x2 ) ≼m (y1 , y2 ) =⇒ g x(1) , x(2) ≥ g y(1) , y(2) . Lemma 3.3 (Karlin and Rinott, 1983). Let X ∈ ℜn have a log-concave density, and let φ(x, a) be convex in (x, a) ∈ ℜn+m . Then g (a) = P (φ(X , a) ≤ t ) is log-concave on ℜm . Lemma 3.4. Let X1 and X2 be independent and symmetric random variables defined on ℜ with each Xi having a log-concave density function fi . If |X1 | ≤lr |X2 |, then, for any a, b ∈ ℜ2+ ,
2 2 a ≽m b =⇒ P a X≥t ≥P b X≥t , i=1 (i) i i=1 (i) i
t ≥ 0,
or, equivalently, 2
a(i) Xi ≤peak
i =1
2
b(i) Xi .
i =1
Proof. Due to the symmetry of X1 and X2 , it holds that P a(1) X1 + a(2) X2 ≤ t = P a(1) X1 + a(2) X2 ≥ −t ,
∀ t ≥ 0.
For any fixed t ≥ 0, we define g (a1 , a2 ) = P (a1 X1 + a2 X2 ≤ t ) . It is sufficient to prove that
a(1) , a(2) ≽m b(1) , b(2) =⇒ g a(1) , a(2) ≤ g b(1) , b(2) .
(3.2)
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S. Ju, X. Pan / Statistics and Probability Letters 114 (2016) 93–98
Note that g is log-concave by Lemma 3.3. Further, we have g a(2) , a(1) − g a(1) , a(2) =
I{a(2) x1 +a(1) x2 ≤t } f1 (x1 )f2 (x2 )dx1 dx2 − I{a(1) x1 +a(2) x2 ≤t } f1 (x1 )f2 (x2 )dx1 dx2 2 ℜ2 ℜ = I{a(2) x1 +a(1) x2 ≤t , x1 ≤x2 } − I{a(1) x1 +a(2) x2 ≤t , x1 ≤x2 } 2 ℜ
× [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2 = ∆1 + ∆2 + ∆3 , where IA is the indicator function of set A ⊂ ℜ2 , and
∆1 = ∆2 =
ℜ2+
I{a(2) x1 +a(1) x2 ≤t , x1 ≤x2 } − I{a(1) x1 +a(2) x2 ≤t , x1 ≤x2 } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2 ,
ℜ− ×ℜ+
∆3 =
I{a(2) x1 +a(1) x2 ≤t } − I{a(1) x1 +a(2) x2 ≤t } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2 ,
I{a(2) x1 +a(1) x2 ≤t , x1 ≤x2 } − I{a(1) x1 +a(2) x2 ≤t , x1 ≤x2 } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2 .
ℜ2−
From the definition of |X1 | ≤lr |X2 |, it is known that |X1 | and |X2 | are SAI. By Lemma 2.6, it holds that
∆1 =
ℜ2+
I{a(2) x1 +a(1) x2 ≤t ≤a(1) x1 +a(2) x2 , x1 ≤x2 } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2 ≥ 0,
(3.3)
and
∆2 =
I{−a(2) x1 +a(1) x2 ≤t } − I{−a(1) x1 +a(2) x2 ≤t } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2
I{−a(2) x1 +a(1) x2 ≤t , x1 ≤x2 } − I{−a(1) x1 +a(2) x2 ≤t , x1 ≤x2 } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2
ℜ2+
= ℜ2
+
+ ℜ2+
= ℜ2+
I{−a(1) x2 +a(2) x1 ≤t , x1 ≤x2 } − I{−a(2) x2 +a(1) x1 ≤t , x1 ≤x2 } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2
I{|−a(2) x1 +a(1) x2 |≤t ≤−a(1) x1 +a(2) x2 , x1 ≤x2 } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2
≥ 0.
(3.4)
Note that ∆3 = 0 for any fixed t ≥ 0 and a(1) , a(2) ∈ ℜ2+ . Combining (3.3) and (3.4), it holds that ∆1 + ∆2 + ∆3 ≥ 0, which yields
g a(2) , a(1) ≥ g a(1) , a(2) .
Therefore, g satisfies the conditions in Lemma 3.2, and hence (3.2) holds. This invokes the desired conclusion.
Now, we are ready to present the following result. Theorem 3.5. Let X1 , . . . , Xn be independent and symmetric random variables defined on ℜ with each Xi having a log-concave density function fi . If |X1 | ≤lr |X2 | ≤lr · · · ≤lr |Xn |, then, for any a, b ∈ ℜn+ ,
n n a ≽m b =⇒ P a X≥t ≥P b X≥t , i=1 (i) i i=1 (i) i or, equivalently, n i =1
a(i) Xi ≤peak
n
b(i) Xi .
i=1
Proof. By the nature of majorization, we only need to prove that a(1) X1 + a(2) X2 +
n i=3
ci Xi ≤peak b(1) X1 + b(2) X2 +
n i=3
ci Xi
t ≥ 0,
S. Ju, X. Pan / Statistics and Probability Letters 114 (2016) 93–98
97
holds for all
a(1) , a(2) , c3 , . . . , cn ≽m b(1) , b(2) , c3 , . . . , cn ,
with a(1) , a(2) ≽m b(1) , b(2) . From Lemma 3.4, it follows that
2
a(i) Xi ≤peak
2
i =1
b(i) Xi .
i =1
Using Lemma 3.1, we get the desired result.
Remark. When X1 , . . . , Xn are i.i.d. random variables, Theorem 3.5 recovers the result in Proschan (1965). However, our proof is different from that in the literature. By the similar argument, we have the following result for bivariate SAI random variables. Theorem 3.6. Let X1 and X2 be symmetric random variables defined on ℜ with a log-concave joint density function. If |X1 | and |X2 | are stochastically arrangement increasing, then for all a, b ∈ ℜ2+ ,
2 2 b X≥t , a X≥t ≥P a ≽m b =⇒ P i=1 (i) i i=1 (i) i
t ≥ 0,
or, equivalently, 2
a(i) Xi ≤peak
2
i =1
b(i) Xi .
i =1
Proof. Due to the symmetry of X1 and X2 , it holds that P a(1) X1 + a(2) X2 ≤ t = P a(1) X1 + a(2) X2 ≥ −t ,
∀ t ≥ 0.
For any fixed t ≥ 0, denote g (a1 , a2 ) = P (a1 X1 + a2 X2 ≤ t ) . It is sufficient for us to prove that
a(1) , a(2) ≽m b(1) , b(2) =⇒ g a(1) , a(2) ≤ g b(1) , b(2) .
By Lemma 3.3, it holds that g is log-concave. Using a similar argument to that of Lemma 3.4, we have g a(2) , a(1) ≥ g a(1) , a(2) .
Therefore, the desired result follows from Lemma 3.2.
Remark. It is interesting to study whether Theorem 3.6 is true for SAI random vectors with arbitrary dimensions. However, the problem still remains open. Acknowledgments The author would like to thank the anonymous referee for the constructive comments and suggestions which led us to significantly improve the paper. Besides, the author is grateful to Professor Taizhong Hu and Maochao Xu for their comments and contributions which led us to provide a more readable description on the method and the main results. The author was supported by the NNSF of China (No. 11401558), the Fundamental Research Funds for the Central Universities (No. WK2040160010) and China Postdoctoral Science Foundation (No. 2014M561823). References Birnbaum, Z.W., 1948. On random variables with comparable peakedness. Ann. Math. Statist. 19, 76–81. Boland, P.J., Proschan, F., 1988. Multivariate arrangement increasing functions with applications in probability and statistics. J. Multivariate Anal. 25, 286–298. Cai, J., Wei, W., 2014. Some new notions of dependence with applications in optimal allocation problems. Insurance Math. Econom. 55, 200–209. Cai, J., Wei, W., 2015. Notions of multivariate dependence and their applications in optimal portfolio selections with dependent risks. J. Multivariate Anal. 138, 156–169. Hollander, M., Proschan, F., Sethuraman, J., 1977. Functions decreasing in transposition and their applications in ranking problems. Ann. Statist. 5, 722–733. Ibragimov, R., 2007. Efficiency of linear estimators under heavy-tailedness: convolutions of α -symmetric distributions. Econometric Theory 23, 501–517. Karlin, S., Rinott, Y., 1983. Comparison of measures, multivariate majorization, and applications to statistics. In: Karlin, S., et al. (Eds.), Studies in Econometrics, Time Series, and Multivariate Statistics. Academic Press, New York, pp. 465–489.
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Ma, C., 1998. On peakedness of distributions of convex combinations. J. Statist. Plann. Inference 70, 51–56. Marshall, A.W., Olkin, I., Arnold, B.C., 2011. Inequalities: Theory of Majorization and Its Applications. Springer, New York. Müller, A., Stoyan, D., 2002. Comparison Methods for Stochastic Models and Risks. John Wiley & Sons, Ltd., Chichester. Pan, X., Xu, M., Hu, T., 2013. Some inequalities of linear combinations of independent random variables: II. Bernoulli 19 (5A), 1776–1789. Pan, X., Yuan, M., Kochar, S., 2015. Stochastic comparisons of weighted sums of arrangement increasing random variables. Statist. Probab. Lett. 102, 42–50. Proschan, F., 1965. Peakedness of distributions of convex combinations. Ann. Math. Statist. 36, 1703–1706. Shaked, M., Shanthikumar, J.G., 2007. Stochastic Orders. Springer, New York. Shanthikumar, J.G., Yao, D.D., 1991. Bivariate characterization of some stochastic order relations. Adv. Appl. Probab. 23, 642–659. Tong, Y.L., 1994. Some recent developments on majorization inequalities in probability and statistics. Linear Algebra Appl. 199, 69–90. Xu, M., Hu, T., 2011. Some inequalities of linear combinations of independent random variables: I. J. Appl. Probab. 48, 1179–1188. Zhao, P., Chan, P.S., Ng, H.K.T., 2011. Peakedness for weighted sums of symmetric random variables. J. Statist. Plann. Inference 141, 1737–1743.
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
A new proof for the peakedness of linear combinations of random variables Shan Ju, Xiaoqing Pan ∗ Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
article
info
Article history: Received 8 October 2015 Received in revised form 14 March 2016 Accepted 14 March 2016 Available online 30 March 2016
abstract A new proof is given to compare linear combinations of independent and possibly non-identically distributed random variables, as well as bivariate SAI random variables in the sense of peakedness order. The main results extend those given in Proschan (1965) and Ma (1998). © 2016 Elsevier B.V. All rights reserved.
MSC: 60E15 62N05 62G30 Keywords: Peakedness Majorization Stochastically arrangement increasing Log-concavity Schur-concave [Schur-convex]
1. Introduction Let X1 , . . . , Xn be independent and identically distributed (i.i.d.) random variables with a symmetric log-concave density function. Theorem 2.3 of Proschan (1965) shows that, for any a = (a1 , . . . , an ), b = (b1 , . . . , bn ) ∈ ℜn+ , it holds
n n a ≽m b =⇒ P aX≥t ≥P bX≥t , i=1 i i i =1 i i
t ≥ 0,
where ‘≽m ’ denotes the majorization order (see Section 2 for the formal definition). This topic has been followed by many researchers because of its wide applications in statistics, reliability theory, economic theory and other fields. For example, Tong (1994) studied various applications of this result in probability and statistics. Ma (1998) generalized this result from i.i.d. random variables to independent and possibly non-identically distributed (i.ni.d.) random variables with symmetric log-concave density functions. The application of this result in economic theory was discussed in Ibragimov (2007). Zhao et al. (2011) and Xu and Hu (2011) studied various sufficient conditions for comparing the distributions of linear combinations of independent random variables. One may also refer to Pan et al. (2013, 2015) and references therein for the recent development of this topic.
∗
Corresponding author. E-mail addresses: [email protected] (S. Ju), [email protected] (X. Pan).
http://dx.doi.org/10.1016/j.spl.2016.03.012 0167-7152/© 2016 Elsevier B.V. All rights reserved.
94
S. Ju, X. Pan / Statistics and Probability Letters 114 (2016) 93–98
It is worth mentioning that all the methodologies in the literature for comparing the distributions of linear combinations of random variables are adopted from that in Proschan (1965). However, the methodology of Proschan (1965) cannot be applied to the case of dependent random variables. In this note, we propose a new proof for Theorem 2.3 of Proschan (1965). The new approach can be used not only for the case of i.ni.d. random variables but also for the case of dependent variables. The rest of the paper is organized as follows. Section 2 recalls the definitions of some stochastic orders, majorization, and stochastic arrangement increasing random vectors. The main results are presented in Section 3. Throughout this paper, the terms increasing and decreasing stand for non-decreasing and non-increasing, respectively, and all expectations are implicitly assumed to be finite whenever they appear. 2. Preliminaries In this section, we recall some notions which are pertinent to our main discussion. Definition 2.1 (Shaked and Shanthikumar, 2007). Let X and Y be two random variables. X is said to be smaller than Y (i) in the usual stochastic order, denoted by X ≤st Y , if P(X > t ) ≤ P(Y > t ) for all t; (ii) 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, where f and g are the density functions of X and Y , respectively. The above stochastic orders can be used to compare the magnitudes of random variables. The relationship between them is well-known in the literature (see Shaked and Shanthikumar, 2007 and Müller and Stoyan, 2002): X ≤lr Y =⇒ X ≤st Y . Definition 2.2 (Birnbaum, 1948). Let X and Y be random variables and let u and v be real constants. X is said to be more peaked about u than Y about v , denoted by X ≤peak Y , if P (|X − u| ≥ t ) ≤ P (|Y − v| ≥ t ) ,
t ≥ 0.
For u = v = 0, we simply say that X is more peaked than Y , or, equivalently, |X | ≤st |Y |. The majorization deals with a partial ordering of the diversity of the components of a vector. For more details on the theory of majorization, see Marshall et al. (2011). Let x(1) ≤ x(2) ≤ · · · ≤ x(n) be the increasing arrangement of the components of the vector x = (x1 , . . . , xn ). Definition 2.3. For vectors x, y ∈ ℜn , x is said to be majorized by y, denoted by x ≼m y, if j i=1
x(i) ≥
j
y(i)
n
i=1
x(i) =
n
i =1
y(i) and
for j = 1, . . . , n − 1.
i=1
A real-valued function φ defined on a set A ⊆ ℜn is said to be Schur-concave [Schur-convex] on A if, for any x, y ∈ A, x ≽m y =⇒ φ(x) ≤ [≥] φ(y ); and φ is said to be log-concave on A ⊂ ℜn if A is a convex set and, for any x, y ∈ A and α ∈ [0, 1],
φ(α x + (1 − α)y ) ≥ [φ(x)]α [φ(y )]1−α . Denote π = (π (1), . . . , π (n)) a permutation of {1, . . . , n}, and define π (x) = (xπ (1) , . . . , xπ (n) ) for x ∈ ℜn . For any 1 ≤ i ̸= j ≤ n, let (πij (1), . . . , πij (n)) with πij (i) = j, πij (j) = i and πij (k) = k for k ̸∈ {i, j}. Definition 2.4. A real-valued function g (x) defined on ℜn is said to be an arrangement increasing (AI) function if
(xi − xj ) g (x) − g (πij (x)) ≤ 0, for all x ∈ ℜn and any pair (i, j) such that 1 ≤ i < j ≤ n. For more details on AI functions and their applications, one may refer to Hollander et al. (1977), and Boland and Proschan (1988). Recently, Cai and Wei (2014, 2015) introduced the following notion to characterize the monotonicity of mutually dependent random variables. Definition 2.5. A random vector X = (X1 , . . . , Xn ) is said to be stochastic arrangement increasing (SAI) if E[g (X )] ≥ E[g (πij (X ))], for any AI function g and all pair (i, j) such that 1 ≤ i < j ≤ n.
S. Ju, X. Pan / Statistics and Probability Letters 114 (2016) 93–98
95
An absolutely continuous SAI random vector can be characterized by its joint probability density as follows. Lemma 2.6. An absolutely continuous random vector is SAI if and only if the corresponding probability density function is AI. This characterization was pointed out in Shanthikumar and Yao (1991) and further discussed in Cai and Wei (2014), and Pan et al. (2015). The examples of SAI random vector include: (1) X1 , . . . , Xn are i.i.d. random variables; (2) X1 , . . . , Xn are exchangeable random variables; (3) X1 , . . . , Xn are independent random variables satisfying X1 ≤lr X2 ≤lr · · · ≤lr Xn . Many well-known distributions are SAI (Hollander et al., 1977), which include multinomial, negative multinomial, multivariate hypergeometric, dirichlet, inverted dirichlet, negative multivariate hypergeometric, dirichlet compound negative multinomial, multivariate logarithmic series distribution, multivariate F distribution, multivariate Pareto distribution, and multivariate normal distribution. 3. Main results In this section, we present a new proof for Theorem 2.3 of Proschan (1965), and also discuss the case of bivariate SAI random variables. The following lemmas are needed to present our main results. Lemma 3.1 (Birnbaum, 1948 and Shaked and Shanthikumar, 2007). Let X1 , . . . , Xn and Y1 , . . . , Yn be two sets of independent random variables with Xi ’s and Yi ’s (i = 1, . . . , n) having distribution functions that are symmetric about possibly different centers and unimodal densities with possibly some probability mass at their respective centers. If Xi ≤peak Yi for i = 1, . . . , n, then n
Xi ≤peak
i =1
n
Yi .
i=1
Lemma 3.2 (Pan et al., 2013). If g : ℜ2 → ℜ+ is log-concave and −g is AI, i.e. g x(2) , x(1) ≥ g x(1) , x(2)
forall (x1 , x2 ) ∈ ℜ2 ,
(3.1)
then
(x1 , x2 ) ≼m (y1 , y2 ) =⇒ g x(1) , x(2) ≥ g y(1) , y(2) . Lemma 3.3 (Karlin and Rinott, 1983). Let X ∈ ℜn have a log-concave density, and let φ(x, a) be convex in (x, a) ∈ ℜn+m . Then g (a) = P (φ(X , a) ≤ t ) is log-concave on ℜm . Lemma 3.4. Let X1 and X2 be independent and symmetric random variables defined on ℜ with each Xi having a log-concave density function fi . If |X1 | ≤lr |X2 |, then, for any a, b ∈ ℜ2+ ,
2 2 a ≽m b =⇒ P a X≥t ≥P b X≥t , i=1 (i) i i=1 (i) i
t ≥ 0,
or, equivalently, 2
a(i) Xi ≤peak
i =1
2
b(i) Xi .
i =1
Proof. Due to the symmetry of X1 and X2 , it holds that P a(1) X1 + a(2) X2 ≤ t = P a(1) X1 + a(2) X2 ≥ −t ,
∀ t ≥ 0.
For any fixed t ≥ 0, we define g (a1 , a2 ) = P (a1 X1 + a2 X2 ≤ t ) . It is sufficient to prove that
a(1) , a(2) ≽m b(1) , b(2) =⇒ g a(1) , a(2) ≤ g b(1) , b(2) .
(3.2)
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S. Ju, X. Pan / Statistics and Probability Letters 114 (2016) 93–98
Note that g is log-concave by Lemma 3.3. Further, we have g a(2) , a(1) − g a(1) , a(2) =
I{a(2) x1 +a(1) x2 ≤t } f1 (x1 )f2 (x2 )dx1 dx2 − I{a(1) x1 +a(2) x2 ≤t } f1 (x1 )f2 (x2 )dx1 dx2 2 ℜ2 ℜ = I{a(2) x1 +a(1) x2 ≤t , x1 ≤x2 } − I{a(1) x1 +a(2) x2 ≤t , x1 ≤x2 } 2 ℜ
× [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2 = ∆1 + ∆2 + ∆3 , where IA is the indicator function of set A ⊂ ℜ2 , and
∆1 = ∆2 =
ℜ2+
I{a(2) x1 +a(1) x2 ≤t , x1 ≤x2 } − I{a(1) x1 +a(2) x2 ≤t , x1 ≤x2 } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2 ,
ℜ− ×ℜ+
∆3 =
I{a(2) x1 +a(1) x2 ≤t } − I{a(1) x1 +a(2) x2 ≤t } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2 ,
I{a(2) x1 +a(1) x2 ≤t , x1 ≤x2 } − I{a(1) x1 +a(2) x2 ≤t , x1 ≤x2 } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2 .
ℜ2−
From the definition of |X1 | ≤lr |X2 |, it is known that |X1 | and |X2 | are SAI. By Lemma 2.6, it holds that
∆1 =
ℜ2+
I{a(2) x1 +a(1) x2 ≤t ≤a(1) x1 +a(2) x2 , x1 ≤x2 } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2 ≥ 0,
(3.3)
and
∆2 =
I{−a(2) x1 +a(1) x2 ≤t } − I{−a(1) x1 +a(2) x2 ≤t } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2
I{−a(2) x1 +a(1) x2 ≤t , x1 ≤x2 } − I{−a(1) x1 +a(2) x2 ≤t , x1 ≤x2 } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2
ℜ2+
= ℜ2
+
+ ℜ2+
= ℜ2+
I{−a(1) x2 +a(2) x1 ≤t , x1 ≤x2 } − I{−a(2) x2 +a(1) x1 ≤t , x1 ≤x2 } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2
I{|−a(2) x1 +a(1) x2 |≤t ≤−a(1) x1 +a(2) x2 , x1 ≤x2 } [f1 (x1 )f2 (x2 ) − f1 (x2 )f2 (x1 )] dx1 dx2
≥ 0.
(3.4)
Note that ∆3 = 0 for any fixed t ≥ 0 and a(1) , a(2) ∈ ℜ2+ . Combining (3.3) and (3.4), it holds that ∆1 + ∆2 + ∆3 ≥ 0, which yields
g a(2) , a(1) ≥ g a(1) , a(2) .
Therefore, g satisfies the conditions in Lemma 3.2, and hence (3.2) holds. This invokes the desired conclusion.
Now, we are ready to present the following result. Theorem 3.5. Let X1 , . . . , Xn be independent and symmetric random variables defined on ℜ with each Xi having a log-concave density function fi . If |X1 | ≤lr |X2 | ≤lr · · · ≤lr |Xn |, then, for any a, b ∈ ℜn+ ,
n n a ≽m b =⇒ P a X≥t ≥P b X≥t , i=1 (i) i i=1 (i) i or, equivalently, n i =1
a(i) Xi ≤peak
n
b(i) Xi .
i=1
Proof. By the nature of majorization, we only need to prove that a(1) X1 + a(2) X2 +
n i=3
ci Xi ≤peak b(1) X1 + b(2) X2 +
n i=3
ci Xi
t ≥ 0,
S. Ju, X. Pan / Statistics and Probability Letters 114 (2016) 93–98
97
holds for all
a(1) , a(2) , c3 , . . . , cn ≽m b(1) , b(2) , c3 , . . . , cn ,
with a(1) , a(2) ≽m b(1) , b(2) . From Lemma 3.4, it follows that
2
a(i) Xi ≤peak
2
i =1
b(i) Xi .
i =1
Using Lemma 3.1, we get the desired result.
Remark. When X1 , . . . , Xn are i.i.d. random variables, Theorem 3.5 recovers the result in Proschan (1965). However, our proof is different from that in the literature. By the similar argument, we have the following result for bivariate SAI random variables. Theorem 3.6. Let X1 and X2 be symmetric random variables defined on ℜ with a log-concave joint density function. If |X1 | and |X2 | are stochastically arrangement increasing, then for all a, b ∈ ℜ2+ ,
2 2 b X≥t , a X≥t ≥P a ≽m b =⇒ P i=1 (i) i i=1 (i) i
t ≥ 0,
or, equivalently, 2
a(i) Xi ≤peak
2
i =1
b(i) Xi .
i =1
Proof. Due to the symmetry of X1 and X2 , it holds that P a(1) X1 + a(2) X2 ≤ t = P a(1) X1 + a(2) X2 ≥ −t ,
∀ t ≥ 0.
For any fixed t ≥ 0, denote g (a1 , a2 ) = P (a1 X1 + a2 X2 ≤ t ) . It is sufficient for us to prove that
a(1) , a(2) ≽m b(1) , b(2) =⇒ g a(1) , a(2) ≤ g b(1) , b(2) .
By Lemma 3.3, it holds that g is log-concave. Using a similar argument to that of Lemma 3.4, we have g a(2) , a(1) ≥ g a(1) , a(2) .
Therefore, the desired result follows from Lemma 3.2.
Remark. It is interesting to study whether Theorem 3.6 is true for SAI random vectors with arbitrary dimensions. However, the problem still remains open. Acknowledgments The author would like to thank the anonymous referee for the constructive comments and suggestions which led us to significantly improve the paper. Besides, the author is grateful to Professor Taizhong Hu and Maochao Xu for their comments and contributions which led us to provide a more readable description on the method and the main results. The author was supported by the NNSF of China (No. 11401558), the Fundamental Research Funds for the Central Universities (No. WK2040160010) and China Postdoctoral Science Foundation (No. 2014M561823). References Birnbaum, Z.W., 1948. On random variables with comparable peakedness. Ann. Math. Statist. 19, 76–81. Boland, P.J., Proschan, F., 1988. Multivariate arrangement increasing functions with applications in probability and statistics. J. Multivariate Anal. 25, 286–298. Cai, J., Wei, W., 2014. Some new notions of dependence with applications in optimal allocation problems. Insurance Math. Econom. 55, 200–209. Cai, J., Wei, W., 2015. Notions of multivariate dependence and their applications in optimal portfolio selections with dependent risks. J. Multivariate Anal. 138, 156–169. Hollander, M., Proschan, F., Sethuraman, J., 1977. Functions decreasing in transposition and their applications in ranking problems. Ann. Statist. 5, 722–733. Ibragimov, R., 2007. Efficiency of linear estimators under heavy-tailedness: convolutions of α -symmetric distributions. Econometric Theory 23, 501–517. Karlin, S., Rinott, Y., 1983. Comparison of measures, multivariate majorization, and applications to statistics. In: Karlin, S., et al. (Eds.), Studies in Econometrics, Time Series, and Multivariate Statistics. Academic Press, New York, pp. 465–489.
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