ISRN Applied MathematicsVolume 2013 (2013), Article ID 780587, 5 pageshttp://dx.doi.org/10.1155/2013/780587

Research Article

On the Dispersive Ordering and Applications

Tran Loc Hung¹ and Nguyen Van Son²

¹Department of Mathematics and Statistics, Faculty of Basic Science, University of Finance-Marketing (UFM), 306 Nguyen Trong Tuyen Street, Tan Binh District, Ho Chi Minh City, Vietnam²Department of Mathematics, Hue University of Science, 77 Nguyen Hue Street, Hue City, Vietnam

Received 20 May 2013; Accepted 24 July 2013

Academic Editors: M. Brünig and Q. Song

Copyright © 2013 Tran Loc Hung and Nguyen Van Son. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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

1. Introduction

Let be a random variable defined on a probability space , with distribution and mean . A random variable is called to belong to class , if its mean is finite. From now on, denotes the dispersion function of -random 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 -norm and it can be considered as a generalization of the mean absolute deviation and the median absolute deviation 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, 7–9] 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 -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 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 -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 -norm.

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.

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 , , and stand for the convergence in distribution, convergence in probability, and convergence in -norm, respectively. For the convenience of the reader we repeat the relevant material from [2, 3] without proofs. Specifically, we have for every 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 -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 be a sequence of -random variables. If there exists , such that  and if  then , for all .(5)Let be a sequence of -random variables, and let be the corresponding of dispersion functions. Suppose that as , for every . Then, we have the following.(a) is a convex function on . (b)Let be the set of all points , such that , , and exist. Then is a dense set in and (c)Let be the set of all points such that exists. Then for , (6)Let , and let be the corresponding dispersion functions. Assume that as , for every . Then,(a)(b)(7)Let be a sequence of -random variables, and let be the corresponding sequence of dispersion functions. If the assumption holds and as , for every , then there exists a distribution function such that  (see [4] for more details).

3. Main Results

In this section, all the random variables or probability distribution functions mentioned are related to space.

According to the results from Muñoz-Perez and Sanchez-Gomez [2, 3], for , we say that the random variable is at least as dispersed as , denoted by or , if

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 Hence, where Besides, Therefore Using the results just obtained, we have By an argument analogous to the previous one, we get We infer that This completes the proof of the lemma.

Theorem 2. Suppose that the random variables . Then, we have the following. (1)If and , then .(2)If and , then . That means that (3)If , then .(4)Let , , and be independent random variables in and . Then,

Proof. Properties and are obvious.For the property , suppose that . We will show that , since the rest is equivalent to this.Note that According to that, the dispersion function is a convex function whose derivative exists almost everywhere and is bounded by and . Hence, This gives Besides, Combining the (27) and (28) or (29), we get the complete proof.The last property can be obtained from the previous lemma.

The following theorem gives us an important property of dispersive ordering.

Theorem 3. Let be a sequence of distribution functions. If they are monotone and bounded in the meaning of dispersive ordering, then they converge weakly.

Proof. Let be the corresponding dispersion functions of . Since is monotone and bounded, the sequence is monotone and bounded for each as well. That means that there exists such that for every .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 is a martingale and -bounded, then the corresponding sequence of distribution functions converges weakly.

Proof. It is know that if is a martingale, then is a nondecreasing sequence. From the bounded condition and Theorem 3, we obtain the conclusion.

Theorem 5. Let be a sequence of -independent random variables. Moreover, suppose that is a -random variable , satisfying Then where .

Proof. Without loss of generality, we assume that , for all .Suppose that is a sequence of independent random variables such that , for all , .We have where is the notation of convolution between distribution functions and Note that We obtain Since , we have This means that It is implied from (35) and (37) that According to the previous proposition, Otherwise, Hence, 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 -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 is a random sample from the family of distribution and the estimator is called a minimum-dispersive unbiased estimator if is unbiased and holds for every unbiased estimator of . It can be shown that the sample mean is a good estimator in the meaning of dispersive ordering.

Proposition 7. If is a random sample from the family of distribution with finite mean, then where , .

Proof. Since , , , have the same distribution, we have and this implies the proof.

Proposition 8. If then is a consistent estimator of , where is the dispersion function of degenerated random variable at .

Proof. According to the properties of dispersion functions, we have , and it follows that The proof is completed.

Acknowledgments

The authors wish to express their gratitude to the referees for valuable remarks and comments improving the previous version of this paper. This work is supported by Vietnam's National Foundation For Science and Technology Development (NAFOSTED, Vietnam), Grant 101.01-2010.02.

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