# Discussion Paper

## Abstract

This paper develops some new stochastic dominance (SD) rules for ranking transformations on a random variable, which is the first time to study ranking approach for transformations on the discrete framework. By using the expected utility theory, the authors first present a sufficient condition for general transformations by first degree SD (FSD), and further develop it into the necessary and sufficient condition for the monotonic transformations. For the second degree SD (SSD) case, the authors divide the monotonic transformations into increasing and decreasing ones, and respectively derive the necessary and sufficient conditions for the two situations. For two different discrete random variables with the same possible states, they obtain the sufficient and necessary condition for FSD and SSD, respectively. The feature of the new SD rules is that each FSD condition is represented by the transformation functions and each SSD condition is characterized by the transformation functions and the probability distributions of the random variable. This is different from the existing SD approach where they are described by cumulative distribution functions. In this way, the authors construct a new theoretical paradigm for transformations on the discrete random variable. Finally, a numerical example is provided to show the effectiveness of the new SD rules.

## JEL Classification

## Cite As

## Assessment

# Comments and Questions

The authors address an interesting issue. Just as the authors say, economic and financial activities usually induce transformations of an initial risk, which results in a new problem of how to rank transformations on the same random variable. It is advantageous to adapt the stochastic dominance ranking procedure to include ...[more]

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the ranking of transformations of random variables. This is the case whenever the emphasis in the analysis is on pairs of random variables which are related to one another by means of a transformation, or on sets of random variables where each are related to a common random variable by means of a transformation.

I believe the paper has several innovative ideas. The authors generalize Meyer’s SD criteria from the following aspects: replacing transformations on continuous random variables by those of discrete random variables, considering both increasing and decreasing transformations and greatly relaxing the restrictions of transformations. I also notice that the SD rules developed in the paper can also be applied to rank the ordinary discrete random variables (not transformations) in some case (see Theorem 7).

On the basis of above considerations, I recommend it can be published quickly in Economics E-Journal. However, I will present an advice.

(1) In the proof of Theorem 2 (see section 3). Step 2 is not needed and it can be replaced by the existing conclusion that X dominates by Y by FSD if and only if -Y dominates -X by FSD (see Shaked and Shanthikumar (2007)).

(2) In section 5. Indeed, Theorem 7 extends the FSD algorithm and SSD algorithm (see Levy (2006), section 5) from equally possible to more general case.

Thank for your comments. And I think it brings me very useful suggestions.

(1) I have reviewed the literature Shaked and Shanthikumar (2007) and find the conclusion that dominates by FSD if and only if dominates by FSD. I will shorten the
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proof of Theorem 2 with this result in the future version of the paper.

(2) I have carefully read Chapter 5 of Levy (2006), pp. 173-182 and find that Levy (2006) provides the FSD and SSD algorithms for equal-probability distributed and discrete random variable, while Theorem 7 further extend these two algorithms to more general discrete random variables only if they have the same possible states.

Report to “A New Approach of Stochastic Dominance for Ranking Transformations on the Discrete Random Variable"

The authors put forward a choice model expressed by transformations. It is well-known that the SD approach is one of the most important choice criteria, and the authors adapt the
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SD approach to include the ranking of transformations. Based on the expected utility theory, FSD and SSD for transformations on discrete random variables are proposed, which can rank transformations directly by the transformation functions and the probability distribution of the original random variable. Compared with the classical SD approach, the new SD rules are more concise.

This paper addresses a meaningful issue and gets some good results. Theorem 1, Theorem 3 and Theorem 5 give the sufficient conditions for one transformation dominating another by SD rules. Theorem 2, Theorem 4 and Theorem 6 further develop the above conditions into the necessary and sufficient conditions. Theorem 7 shows that the new SD rules can rank not only the transformations on a discrete random variable, but also different random variables with the same possible states. I have examined all these SD rules and I believe all of the proofs are rigorous and the approach developed in the paper can be applied to rank transformations resulting from economic, financial, insurance and other areas.

I believe there are some real innovations in this paper and the paper should be published in Economics E-Journal. However, I must point out that there are some unclear expressions in the paper. There are 27 expressions of “the existing SD rules” in the whole paper. Some of it mean the classical SD rules, some of it mean Meyer’s SD rules, and some of it mean both of the above two aspects. I suggest the authors should discrinate these cases.

Thank you for approving my job. I agree with you that “the existing SD rules” is an unclear expression, and some of them are inaccurate. For these 27 expressions, I will check them one by one and correct the inaccuracies.

The authors present some new stochastic dominance criteria for ranking transformations resulting from insurance. And this method is applied to the discrete system for the first time. The authors first construct a choice model about insurance, and point out that the classical stochastic dominance cannot be used directly. Then, they ...[more]

...
propose new FSD and SSD criteria for transformations on discrete random variables. Finally, they provide a numerical example to show the effectiveness of the new SD criteria.

I believe there are some real innovations in this paper. First, the new SD criteria are expressed by the transformation functions and probability mass function, other than the cumulative distribution functions in the classical stochastic dominance. Second, compared with Meyers’s results, this paper propose the SD criteria for transformations on discrete random variables, while Meyer’s results can only be applied to continuous random variables. Finally, the SD criteria developed in the paper greatly lowers the restrictions of transformations, so it enlarge the scope of application for stochastic dominance.

In my opinion the paper is interesting and it can be published in Economics E-Journal. Nevertheless, I will propose some suggestions.

(1) In section 2. The authors introduce the definition of n-th degree stochastic dominance. However, only FSD and SSD are extended to transformations. Then, only FSD and SSD shuld be available in Definition 1.

(2) In section 5. The authors derive new criteria for SD, but not new SD rules.

Thank you for approving my job. After serious considerations, I give the reply to the two suggestions in your comment.

(1) About the first suggestion of Definition 1. It is better to get the higer SD criteria for transformations. Following by Chapter 4 of Levy (2006), I can deduce
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that there are no such criteria for TSD and higher SD. However, it is difficult to prove this claim. I must point out that both Meyer (1989) and Levy (1992) only derive the FSD and SSD criteria. To avoid controversy, this claim is not available in the paper.

One the other hand, the definition of SD is usually introduced as a whole, from FSD to n-th degree stochastic dominance in recent literature about SD method. Certainly, FSD and SSD own an overwhelming position in SD research.

Based on the above considerations, I have added the following sentence:

Considering that FSD and SSD have more practical implication than higher degree SD rules, this paper will focus on FSD and SSD rules in the remaining part of the paper.

(2) About the second suggestion in section 5. I agree with you that “criteria” is more appropriate than “rules”.

The paper addresses an interesting topic. There are many situations where one has to compare two different payoff structures which all depend on the outcome of the same random variable. As an example, in finance applications one could have different structured products written on the same underlying and the question ...[more]

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is whether one of those products is dominated by another.

The authors first construct a choice model resulting from insurance and want to resolve the problem with the stochastic dominance apprach. As noted, the classical stochastic dominace are based on the framework of cumulative distribution functions, and Meyer’s stochastic dominance criteria are only suitable for continuous random variabls, so the authors develop the new SD criteria specially for discrete framework. The new SD approach is proved with serious mathematical proofs.

I believe the paper has some innovative results. In my opinion this paper is interesting and I recommend it can be published as quickly as possible in Economics E-Journal. However, I have some remarks.

● In section 2. Definition 1 introduces the definition of n-th degree stochastic dominance. Since only FSD and SSD are extended to transformations, the last item of Definition 1 is not needed.

● Although the proof of Theorem 2 is rigorous, I think it should be more concise.

Thank you for approving my job. And I will answer your questions in the following.

(1) The first question is similar to the question by Jinfeng Sun. And I give the explaination in the reply to Jinfeng Sun.

(2) The same question is put forward in comment 2. And I
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... will shorten the proof of Theorem 2 by adding the literature of Shaked and Shanthikumar (2007). For more details, see my reply to Comment 2.

Report (Zhijie Zhao)

As the authors point out in the paper, economic and financial activities always induce transformations of an initial risk, which results in a new type of problem of how to rank transformations on the same random variable. I believe address an meaning issue.

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Since the classical SD approach cannot be used directly to rank transformations, the authors develop the new FSD and SSD criteria for ranking transformations on a discrete random variable, which is the first time to consider the ranking approach for transformations on the discrete system. Without the tedious computation of the transformations’ CDFs and their integral, the new SD criteria are more efficient to rank transformations than the classical SD rules. In this sense, the newly established theoretical paradigm, so far as ranking transformations on the discrete random variable, can be viewed as a great substitute for the framework of the cumulative distribution functions in the classical SD approach.

On the basis of above considerations, the paper reaches the standard of a publishable paper in Economics E-Journal. In addition, I think it will be better by introducing the FSD and SSD algorithms in Levy (2006). In fact, Levy (2006) provides the FSD and SSD algorithms for equal-probability distributed and discrete random variable, Theorem 7 further extend these two algorithms to more general discrete random variables only if they have the same possible states.

Thank you for approving my job. Your advice in the comment that “it will be better by introducing the FSD and SSD algorithms in Levy (2006). In fact, Levy (2006) provides the FSD and SSD algorithms for equal-probability distributed and discrete random variable, Theorem 7 further extends these two algorithms ...[more]

... to more general discrete random variables only if they have the same possible states”. I believe it is a useful suggestion and I will adopt it in the future version of the paper.

The authors present some new stochastic dominance criteria for ranking transformations resulting from insurance. And this method is applied to the discrete system for the first time. The authors first construct a choice model about insurance, and point out that the classical stochastic dominance cannot be used directly. Then, they ...[more]

...
propose new FSD and SSD criteria for transformations on discrete random variables. Finally, they provide a numerical example to show the effectiveness of the new SD criteria.

I believe there are some real innovations in this paper. First, the new SD criteria are expressed by the transformation functions and probability mass function, other than the cumulative distribution functions in the classical stochastic dominance. Second, compared with Meyers’s results, this paper propose the SD criteria for transformations on discrete random variables, while Meyer’s results can only be applied to continuous random variables. Finally, the SD criteria developed in the paper greatly lowers the restrictions of transformations, so it enlarge the scope of application for stochastic dominance.

In my opinion the paper is interesting and it can be published in Economics. Nevertheless, I will propose some suggestions.

(1) In section 2. The authors introduce the definition of n-th degree stochastic dominance. However, only FSD and SSD are extended to transformations. Then, only FSD and SSD shuld be available in Definition 1.

(2) In section 5. The authors derive new criteria for SD, but not new SD rules.

I apologize for my wrong operation in adding the comments. I approve the comment of Jinfeng Sun and I will further give my own opinions.

As the authors point out that the classical stochastic dominance is inefficient to rank transformations on random variables. Meyer (1989) proposes the SD criteria for
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transformations on continuous random variables, the paper derive the SD criteria for transformations in discrete framework. I have examined all the theorems and corollaries in the paper and believe the new method has a good application foreground.

In consideration of the paper further enrich the stochastic dominance theory and extends the application range, I recommend it can be published quickly in Economics. However, I have some remarks.

(1) In section 5. I believe there are some correlations between Theorem in the paper and the FSD and SSD algorithms in Chapter 5 of Levy (2006).

(2) In section 3. If we compare the proof of Theorem 2 in the discussion paper with the proof of Theorem 2 from Meyer (1989), then we can see that the latter is much shorter.

I have already answered the questions of Jinfeng Sun. If you are interested in those questions, you can refer to my reply to Jinfeng Sun. About your own questions, I will give you a detailed answer.

(1) Thank you for pointing out the correlations between Theorem 7 in the paper
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and the FSD and SSD algorithms in Chapter 5 of Levy (2006). Actually, Levy (2006) provides the FSD and SSD algorithms for equal-probability distributed and discrete random variable, while Theorem 7 further extend these two algorithms to more general discrete random variables only if they have the same possible states. Such remarks will be added in the future version of the paper.

(2) Just as you point out in the comment, the proof of Theorem 2 is little tedious. And fortunately, with the result that dominates by FSD if and only if dominates by FSD (see Shaked and Shanthikumar , 2007), the proof of Theorem 2 can be greatly reduced.

Unfortunately, I don’t think the proof of Theorem 3 can be proved as briefness as that of Meyer (1989). Since the transformations in Meyer (1989) are assumed to be increasing, continuous, and piecewise differentiable, and Taylor Theorem can be applied in the continuous system. When it comes to the discrete system, there is no such simple tool as Taylor's expansion.

The paper addresses an interesting topic. In many cases, the decision makers should rank transformations resulting from economic and financial activities. It is advantageous to adapt the stochastic dominance ranking procedure to include the ranking of transformations of random variables. This is the case whenever the emphasis in the analysis ...[more]

...
is on pairs of random variables which are related to one another by means of a transformation, or on sets of random variables where each are related to a common random variable by means of a transformation.

Similar to Meyer (1989), the authors propose the FSD and SSD criteria for transformations on discrete random variables. Compared with Meyer (1989) and Levy (1992), there are three innovations in this paper.

--SD criteria for transformations are first extablished to rank transformations on discrete random variables.

--SD criteria for decreasing transformations are proposed, while they are ignored in Meyer (1989) and Levy (1992).

--Restrictions of transformations are great reduced in the paper, whick implies the new SD criteria can be applied to more general case.

In my opinion the paper is interesting and it can be published in Economics.

The paper discusses a meaningful issue and gets some useful results. I believe the approach developed in the paper can be applied to many fields if the emphasis in the analysis is on pairs of random variables which are related to a common random variable by means of a transformation. ...[more]

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This type of relationship often arises when instruments such as insurance or put or call options are modeled.

All the results are proved with serious mathematical proofs. And I believe this new approach can be applied to rank transformations resulting from economic, financial, insurance and other areas. So, I recommend it can be published quickly in Economics.

However, there are some small mistakes should be modified.

1. The first paragraph of section 1. The sentence “Stochastic dominance (SD) is the most famous approaches to compare pairs of prospects” could be revised as “…is one of the most famous…”.

2. The first paragraph of section 1. In the sentence of “Unfortunately, SD approach for ranking random variables is inefficient to …”, the four words “for ranking random variables” are unnecessary.

Thank you for your good suggestion! I will agree with your proposals.

Economic and financial activities in most cases will induce transformations of an initial risk, which results in a new problem of how to rank the transformed random variables. For this issue, the authors develop some new stochastic criteria, which are used to rank transformations of discrete random variables.

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The main contribution of the new approach in the article lies in the following three aspects.

(1) Compared with the classical SD approach, the new SD criteria developed in the article are expressed by the transformation functions and the probability function of the original random variable. Thus, it could be used directly to rank transformations, and the cumulative distribution functions of transformed random variables are not needed in this case.

(2) Compared with Meyer and Levy’s result, the SD criteria developed in the paper are applied to rank transformations on discrete random variables. Then, the approach in the article can be regarded as a natural generalization of Meyer and Levy’s results.

(3) Both the increasing and decreasing transformations are considered in the article, and the stronger restrictions of transformations are ruled out, which further improve and develop Meyer and Levy’s results.

Based on the above considerations, I recommend it can be published in Economics

Report on paper: A New Approach of Stochastic Dominance for Ranking Transformations on the Discrete Random Variable

The authors pose an interesting issue. It is well-known that all human activities, especially in the economic and finical areas, induce transformations of an initial risk. Then an interesting question occurs: how
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to rank such transformations? Although the SD approach is one of the most important tools in ranking distibutions, it cannot be used directly to rank transformations. The article reslove such choice problem by revising the SD rules by replacing cumulative distribution functions with transformaton functions and the probability mass function.

The innovations of this paper are follows. First, this paper construct a theoretical model for ranking transformations of discrete random variables. To the best of my knowledge, this is the first time that the paper employ such approach to discriminate transformation in discrete system. Second, the approach developed in the paper is the meaningful extension of Meyer’s result. The authors not only consider the decreasing transformations, but also great lower the limitations of the transformation functions. It's worth mentioning that the SSD rules for transformations (Theorem3—Theorem 6) are proved with religious mathematic deduction under weaker conditions.

All things considered, I recommend it can be published in Economics E-Journal.

The approaches for ranking transforms of continuous random variables by Stochastic Dominance (SD) have been well presented in the literature. The SD in this paper includes the first degree SD (also known as the stochastic ordering) and the second degree SD (also known as the stop loss ordering ...[more]

... in risk management and actuarial science). The transform of a random loss X, denoted as m(X), can be regarded as the retained amount by an insurer under a reinsurance contract, while X-m(X) can be regarded as the ceded amount to the reinsurer under the same reinsurance contract.

In this paper, the authors discuss the sufficient conditions and sufficient and necessary conditions under which two transforms m(X) and n(X) on a discrete random variable X can be ordered by FSD and SSD. The existing sufficient conditions and sufficient and necessary conditions for ordering m(X) and n(X) by FSD and SSD when X is a continuous random variable normally do not apply to the case when X is a discrete random variable. Furthermore, the new conditions avoid using the cumulative distribution function (CDF) of X as normally there is no closed form expression for its CDF if X is discrete.

Thanks for your comments. I agree that FSD and SSD can be regarded as the stochastic ordering and the stop loss ordering in risk management and actuarial science. The SD approach is used in decision theory and decision analysis to refer to situations where one distribution of outcome can be ...[more]

... ranked as superior to another distribution. While the stochastic ordering and the stop loss ordering is usually used in actuarial science, which is used to compare the distributions of random losses.

Just as you say, transformations of random losses can be regarded as the retained amount by an insurer under a reinsurance contract or the ceded amount to the reinsurer under the same reinsurance contract. Proceeding from this angle, I believe the SD criteria for transformations developed can also be applied to the optimal reinsurance.