On the Dispersive Ordering and Applications

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 function


Introduction
Let  be a random variable defined on a probability space (Ω, A, ), with distribution   and mean ().A random variable  is called to belong to class L 1 , if its mean is finite.From now on,   () denotes the dispersion function of L 1random variable  at a point  ∈ (−∞, +∞), defined as follows: It is to be noticed that the dispersion function in (1) also is known as the absolute deviation function of  at a point  ∈ (−∞, +∞) (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 function   () of  at a point  ∈ (−∞, +∞) has attracted much attention as a dispersion measure of  in L 1 -norm and it can be considered as a generalization of the mean absolute deviation  1 () := |−| and the median absolute deviation  2 () := | − | of a random variable  when  and  exist and are unique; here and throughout this paper  and  denote the mean and median of random variable , 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][8][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 L 1random 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   () of random variable  at a point  ∈ (−∞, +∞) has attracted much attention as a dispersion measure of a random variable  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 the L 1 -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 L 1 -norm.

ISRN Applied Mathematics
The organization of this paper is as follows.In Section 2 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.

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][3][4].Throughout this paper the symbols   →,   →, and → stand for the convergence in distribution, convergence in probability, and convergence in L 1 -norm, respectively.For the convenience of the reader we repeat the relevant material from [2,3] without proofs.Specifically, we have for every  ∈ L 1 the following.
(1) The expression of the distribution   of  and the derivative    () of the dispersion function is defined as follows: where   is a set of continuity points of   .(2) The dispersion function   () of  at a point  ∈ (−∞, +∞) is L 1 -distance between   and   : where   is the distribution function of the degenerate variable at the point .
(3) The equivalent formulae of (1) are or 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 {  ,  ≥ 1} be a sequence of L 1 -random variables.If there exists  > 1, such that sup and if then    () →   (), for all  ∈ (−∞, +∞).
(c) Let H 0 be the set of all points such that   () exists.Then for  ∈ H 0 , lim (see [4] for more details).

Main Results
In this section, all the random variables or probability distribution functions mentioned are related to L 1 space.
According to the results from Muñoz-Perez and Sanchez-Gomez [2,3], for ,  ∈ L 1 , we say that the random variable  is at least as dispersed as , denoted by 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 that  and  are two independent random variables.Then, where   and   are distribution functions of  and , respectively.
Proof.Let  be the distribution function of  + ; we have where Besides, Using the results just obtained, we have  ( This completes the proof of the lemma.
Theorem 2. Suppose that the random variables , ,  ∈ L 1 .Then, we have the following.
(  According to that, the dispersion function is a convex function whose derivative exists almost everywhere and is bounded by −1 and 1.Hence, This gives Besides, 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 (  ) ∈N be a sequence of distribution functions.If they are monotone and bounded in the meaning of dispersive ordering, then they converge weakly.
According to the properties of the dispersion functions shown in Section 2, we get the complete proof.
From the Theorem 3, we have the following interesting results.

Corollary 4.
If {  , F  } is a martingale and L 1 -bounded, then the corresponding sequence of distribution functions converges weakly.
Proof.It is know that if {  , F  } is a martingale, then (|  − |)  is a nondecreasing sequence.From the bounded condition and Theorem 3, we obtain the conclusion.Theorem 5. Let {  ,  ≥ 1} be a sequence of L 1 -independent random variables.Moreover, suppose that  is a L 1 -random variable , satisfying    ≤ , ∀ ∈ N. (30) Then where Proof.Without loss of generality, we assume that () = (  ) = 0, for all  ≥ 1.
We have where * is the notation of convolution between distribution functions and 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 the L 1 -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 ( 1 ,  2 , . . .,   ) is a random sample from the family of distribution (, ) and the estimator θ is called a minimum-dispersive unbiased estimator if θ is unbiased and holds θ  ≤  * for every unbiased estimator  * of .
It can be shown that the sample mean is a good estimator in the meaning of dispersive ordering.
The proof is completed.