# On the Dispersive Ordering and Applications

Authors: Tran Loc Hung; Nguyen Van Son
Journal: ISRN Applied Mathematics
 (2013)

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## Abstract

The purpose of this paper is to present some results related to the dispersive ordering of probability distributions via dispersion functions of theℒ1-random variables. A new approach to the Laws of Large Numbers in ℒ1-norm can be applied via received results. A new concept on minimum-dispersive unbiased estimator is considered, too.


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## Body

## 1. Introduction

LetX be a random variable defined on a probability space (Ω,𝔄,P), with distribution FX and mean E(X). A random variable X is called to belong to class ℒ1, if its mean is finite. From now on, DX(u) denotes the dispersion function of ℒ1-random variable X at a point u∈(-∞,+∞), defined as follows:
(1)DX(u)=E|X-u|=∫-∞+∞‍|x-u|dFX(x),u∈(-∞,+∞).
It is to be noticed that the dispersion function in (1) also is known as the absolute deviation function of X at a point u∈(-∞,+∞) (see [1] for more details). Up to the present, some results related to dispersion functions in term of (1) have been investigated by Muñoz-Perez and Sanchez-Gomez [2, 3], Pham-Gia and Hung [1], and Hung and Son [4]. Also note that the dispersive functions and stochastic ordering have been considered in various papers and they are effective tools in many areas of probability and statistics. Such areas include reliability theory, queuing theory, survival analysis, biology, economics, insurance, actuarial science, operations research, and management science (we refer to [5, 6] for a complete treatment of the problem).It is worth pointing out that the dispersion functionDX(u) of X at a point u∈(-∞,+∞) has attracted much attention as a dispersion measure of X in ℒ1-norm and it can be considered as a generalization of the mean absolute deviation δ1(μ):=E|X-μ| and the median absolute deviation δ2(X):=E|X-Md| of a random variable X when μ and Md exist and are unique; here and throughout this paper μ and Md denote the mean and median of random variable X, respectively. The mean absolute deviation and median absolute deviation that play particular roles in Applied Statistics and Economics have been investigated by Pham-Gia et al. (we refer the reader to [1, 7–9] for deeper discussions).The dispersion function as was stated above is convex and almost everywhere differentiable, and its derivative has most a countable numbers of discontinuity points (see for instance [2, 3]). Lately, some interesting results concerning the connections of the weak convergence of a sequence of ℒ1-random variables with the convergence of their corresponding dispersion functions have been investigated by Hung and Son (see [4] for more details). Thus, the dispersion function DX(u) of random variable X at a point u∈(-∞,+∞) has attracted much attention as a dispersion measure of a random variable X in various problems related to limit theorems of probability theory, applied statistics, and economics.The main aim of this paper is to present some results related to the dispersive ordering of probability distributions via dispersion functions of theℒ1-random variables. The received results are extensions of authors studies in [4], and they are showing a new approach to the Laws of Large Numbers in ℒ1-norm.The organization of this paper is as follows. In Section2 we will recall the main properties of the dispersion functions that will play fundamental roles in the study of next section. For more details about the proofs of results in this section, we refer the reader to Muñoz-Perez and Sanchez-Gomez [2, 3] and Hung and Son [4]. The last section gives some main results on dispersive ordering of probability distributions via dispersion functions and applications.


## 2. Preliminary Results

Some properties of the dispersion functions have been investigated so far; they can be listed as follows. For more details we refer the reader to [2–4]. Throughout this paper the symbols →d, →P, and →ℒ1 stand for the convergence in distribution, convergence in probability, and convergence in ℒ1-norm, respectively. For the convenience of the reader we repeat the relevant material from [2, 3] without proofs. Specifically, we have for every X∈ℒ1 the following.(1)
The expression of the distributionFX of X and the derivative DX′(u) of the dispersion function is defined as follows:
(2)FX(u)=12(DX′(u)+1),∀u∈CF,whereCF is a set of continuity points of FX.(2)
The dispersion functionDX(u) of X at a point u∈(-∞,+∞) is ℒ1-distance between FX and Fu:
(3)DX(u)=∫-∞+∞‍|FX(x)-Fu(x)|dx,whereFu is the distribution function of the degenerate variable at the point u.(3)
The equivalent formulae of (1) are
(4)DX(u)=u-E(X)+2∫u+∞‍(x-u)dFX(x)or(5)DX(u)=E(X)-u+2∫-∞u‍(u-x)dFX(x).On the other hand, Hung and Son in [4] have established the connections of the weak convergence of the random variables with the convergence of the dispersion functions as follows.(4)
Let{Xn,n≥1} be a sequence of ℒ1-random variables. If there exists p>1, such that
(6)supn∈ℕE|Xn|p<+∞,and if(7)Xn⟶dX,asn⟶∞,thenDXn(u)→DX(u), for all u∈(-∞,+∞).(5)
Let{Xn,n≥1} be a sequence of ℒ1-random variables, and let {Dn,n≥1} be the corresponding of dispersion functions. Suppose that Dn(u)→D(u) as n→∞, for every u∈(-∞,+∞). Then, we have the following.
(a)
D(u) is a convex function on (-∞,+∞).(b)
Letℋ be the set of all points u∈(-∞,+∞), such that D′(u), Dn′(u), and n∈ℕ exist. Then ℋ is a dense set in (-∞,+∞) and
(8)limn→∞Dn′(u)=D′(u),∀u∈ℋ.(c)
Letℋ0 be the set of all points such that D′(u) exists. Then for u∈ℋ0,
(9)limu→-∞D′(u)=-1,limu→+∞D′(u)=1.(6)
LetX,X1,…,Xn…∈ℒ1, and let D,D1,…,Dn,… be the corresponding dispersion functions. Assume that Dn(u)→D(u) as n→∞, for every u∈(-∞,+∞). Then,
(a)
(10)Xn⟶dX,asn⟶∞,(b)
(11)limn→∞supu∈ℝ|Dn(u)-D(u)|=0.(7)
Let{Xn,n≥1} be a sequence of ℒ1-random variables, and let {Dn(u),n≥1} be the corresponding sequence of dispersion functions. If the assumption Xn→dX holds and Dn(u)→D(u) as n→∞, for every u∈(-∞,+∞), then there exists a distribution function FX such that
(12)D(u)=∫-∞+∞‍|x-u|dFX(x)(see [4] for more details).


## 3. Main Results

In this section, all the random variables or probability distribution functions mentioned are related toℒ1 space.According to the results from Muñoz-Perez and Sanchez-Gomez [2, 3], for X,Y∈ℒ1, we say that the random variable Y is at least as dispersed as X, denoted by X≤dY or FX≤dFY, if
(13)DX-EX(u)≤DY-EY(u),∀u∈(-∞,+∞).It can be easily seen that a degenerate variables is the lower bound of the family of finite-mean random variables. Before stating the main results of this paper, we first study some properties of dispersive ordering.Lemma 1.
Suppose thatX and Y are two independent random variables. Then,
(14)DX+Y(u)=∫-∞+∞‍DX(u-y)dFY(y)=∫-∞+∞‍DY(u-x)dFY(x),
where FX and FY are distribution functions of X and Y, respectively.Proof.
LetF be the distribution function of X+Y; we have
(15)DX+Y(u)=∫-∞u‍(u-x)dF(x)+∫u+∞‍(x-u)dF(x),F(x)=∫-∞+∞‍FX(x-y)dFY(y).
Hence,
(16)∫au‍(u-x)dF(x)=limn→∞∑k=1kn(u-xnk)F[xnk;xnk+1)=limn→∞∫-∞+∞‍∑k=1kn(u-xnk)FX×[xnk-y;xnk+1-y)dFY(y)=limn→∞∫-∞+∞∑k=1kn[(u-y)-(xnk-y)]FX‍×[xnk-y;xnk+1-y)dFY(y),
where
(17)a=xn1<xn2<⋯<xnkn+1=u,limn→∞sup1≤k≤kn(xnk+1-xnk)=0.
Besides,(18)∑k=1kn[(u-y)-(xnk-y)]FX[xnk-y;xnk+1-y)≤u-a,limn→∞∑k=1kn[(u-y)-(xnk-y)]FX[xnk-y;xnk+1-y)=∫a-yu-y‍(u-y-x)dFX(x).
Therefore(19)limn→∞∫-∞+∞‍∑k=1kn[(u-y)-(xnk-y)]FX×[xnk-y;xnk+1-y)dFY(y)=∫-∞+∞‍∫a-yu-y‍(u-y-x)dFX(x)dFY(y).
Using the results just obtained, we have(20)∫-∞u‍(u-x)dF(x)=∫-∞+∞‍∫-∞u-y‍(u-y-x)dFX(x)dFY(y).
By an argument analogous to the previous one, we get(21)∫u+∞‍(x-u)dF(x)=∫-∞+∞‍∫u-y+∞‍(x-(u-y))dFX(x)dFY(y).
We infer that(22)DX+Y(u)=∫-∞+∞‍|x-u|dF(x)=∫-∞+∞‍∫-∞+∞‍|x-(u-y)|dFX(x)dFY(y)=∫-∞+∞‍DX(u-y)dFY(y).
This completes the proof of the lemma.Theorem 2.
Suppose that the random variablesX,Y,Z∈ℒ1. Then, we have the following.(1)
IfX≤dY and Y≤dZ, then X≤dZ.(2)
IfX≤dY and Y≤dX, then Y=dX. That means that
(23)DX-EX(u)=DY-EY(u),∀u∈(-∞,+∞).(3)
If0<α<1<β, then αX≤dX≤dβX.(4)
LetX, Y, and Z be independent random variables in ℒ1 and X≤dY. Then,
(24)X+Z≤dY+Z.Proof.
Properties(1) and (2) are obvious.
For the property(3), suppose that E(X)=0. We will show that αX≤dX, since the rest is equivalent to this.
Note that(25)DαX(u)=E|αX-u|=αE|X-1αu|=αDX(1αu).
According to that, the dispersion function is a convex function whose derivative exists almost everywhere and is bounded by-1 and 1. Hence,
(26)-1≤DX(u)-DX((1/α)u)u-(1/α)u≤1,∀u≠0.
This gives
(27)αDX(u)+u-αu≥αDX(1αu),∀u>0,αDX(u)-u+αu≥αDX(1αu),∀u<0.
Besides,
(28)DX(u)=E|X-u|=u-E(X)+2∫u+∞‍(x-u)dFX(x),(29)DX(u)=E|X-u|DX(u)=E(X)-u+2∫-∞u‍(u-x)dFX(x)≥|u|.
Combining the (27) and (28) or (29), we get the complete proof.
The last property(4) can be obtained from the previous lemma.The following theorem gives us an important property of dispersive ordering.Theorem 3.
Let(Fn)n∈ℕ be a sequence of distribution functions. If they are monotone and bounded in the meaning of dispersive ordering, then they converge weakly.Proof.
Let(Dn)n∈ℕ be the corresponding dispersion functions of (Fn)n∈ℕ. Since (Fn)n∈ℕ is monotone and bounded, the sequence (Dn(u))n∈ℕ is monotone and bounded for each u∈(-∞,+∞) as well. That means that there exists D(u) such that Dn(u)→D(u) for every u∈(-∞,+∞).
According to the properties of the dispersion functions shown in Section2, we get the complete proof.From the Theorem3, we have the following interesting results.Corollary 4.
If{Xn,ℱn} is a martingale and ℒ1-bounded, then the corresponding sequence of distribution functions converges weakly.Proof.
It is know that if{Xn,ℱn} is a martingale, then (E|Xn-u|)n is a nondecreasing sequence. From the bounded condition and Theorem 3, we obtain the conclusion.Theorem 5.
Let{Xn,n≥1} be a sequence of ℒ1-independent random variables. Moreover, suppose that X is a ℒ1-random variable X, satisfying
(30)Xn≤dX,∀n∈ℕ.
Then
(31)Sn-E(Sn)n⟶ℒ10,
where Sn=X1+X2+⋯+Xn.Proof.
Without loss of generality, we assume thatE(X)=E(Xn)=0, for all n≥1.
Suppose that(Yn)n is a sequence of independent random variables such that P(Yn<x)=P(X<x)=F(x), for all n∈ℕ, x∈(-∞,+∞).
We have(32)D(Y1+Y2+⋯+Yn)/n(u)=1nDY1+Y2+⋯+Yn(nu)=1n∫-∞+∞‍|x-nu|dFn*(x),
where * is the notation of convolution between distribution functions and
(33)Fn*=F*F*⋯*F(ntimes).
Note that
(34)D(Y1+Y2+⋯+Yn)/n(u)≥|u|,1nDY1+Y2+⋯+Yn(nu)≤E|Y1+Y2+⋯+Ynn|+|u|,limn→∞E|Y1+Y2+⋯+Ynn|=0.
We obtain(35)limn→∞D(Y1+Y2+⋯+Yn)/n(u)=|u|.
SinceXn≤dX, we have
(36)Sn≤dY1+Y2+⋯+Yn.
This means that(37)DSn/n(u)≤D(Y1+Y2+⋯+Yn)/n(u).
It is implied from (35) and (37) that
(38)limDSn/n(u)=|u|.
According to the previous proposition,(39)|u|≤D(Sn-E(Sn))/n(u)≤1n∫-∞+∞‍|x-nE(X)-u|dFXn*(x).
Otherwise,
(40)limn→∞1n∫-∞+∞‍|x-nE(X)-u|dFn*(x)=|u|.
Hence,
(41)limn→∞D(Sn-E(Sn))/n(u)=|u|.
And this completes the proof.Note that it makes sense to consider that the dispersive ordering can be applied to limit theorems in probability (for theℒ1-weak law of large numbers) as a new approach and should be more investigated.Moreover, as we know that a famous estimator is minimum-variance unbiased one. This is based on the existence of variance, which is considered as a measure of dispersion. It is natural to link to an estimator based on dispersive ordering.Definition 6.
Suppose that(X1,X2,…,Xn) is a random sample from the family of distribution F(x,θ) and the estimator θ^ is called a minimum-dispersive unbiased estimator if θ^ is unbiased and holds θ^≤dθ* for every unbiased estimator θ* of θ.
It can be shown that the sample mean is a good estimator in the meaning of dispersive ordering.Proposition 7.
If(X1,X2,…,Xn) is a random sample from the family of distribution F(x,θ) with finite mean, then
(42)X1+X2+⋯+Xnn≤dα1X1+α2X2+⋯+αnXn,
where αi>0, α1+α2+⋯+αn=1.Proof.
Sinceα1X1+α2X2+⋯+αnXn, α2X1+α3X2+⋯+α1Xn, …, αnX1+α1X2+⋯+αn-1Xn have the same distribution, we have
(43)nE|α1X1+α2X2+⋯+αnXn-u|=E|α1X1+α2X2+⋯+αnXn-u|+⋯+E|αnX1+α1X2+⋯+αn-1Xn-u|≥E|(α1+α2+⋯+αn)X1+⋯+(α1+α2+⋯+αn)Xn-nu|,
and this implies the proof.Proposition 8.
If(44)Dθ^n-θ(u)⟶D0(u),asn⟶∞,∀u∈ℝ,
then θ^n is a consistent estimator of θ, where D0 is the dispersion function of degenerated random variable at 0.Proof.
According to the properties of dispersion functions, we haveθ^n-θ→d0, and it follows that
(45)θ^n-θ⟶P0,asn⟶∞.
The proof is completed.



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