Determining Distribution for the Product of Random Variables by Using Copulas

Round 1

Reviewer 1 Report

Main comments:

The paper is generally well-written and the result is (in my opinion) very interesting, although I have to admit that the novelty of the results is quite small / the derivations for the density are rather straightforward, taking into account, for example, the book by Cherubini et al. [3].

Minor comments:

- For "independent" random variables (e.g. corollary 1), the result is very well known. This should not be mentioned as part of the headline ("[...] Independent Random Variables by Using Copulas") and it should not be further stressed in the abstract. The title is also a bit confusing as there is no copula in the independent case!

- Proof. of Theorem 1: some arguments are missing, to ensure that Y_2=0 does not cause problems. The inverse transform is defined by X_2 = Y_1/Y_2 (existence?).

- p.3, top: the weights w_1+w_2=1, not X_1+X_2=1.

- In the abstract, the authors state "the product is strongly affected [..] using copulas from the elliptical family but not [...] from the Archimedean family." Looking at the numerical section, the reason might be the choice of parameters. While rho =0.9 might be an extreme choice for this dependence parameter (Fig. 1), theta=4 might not be that extreme (Fig. 3). I'd suggest to use some dependence measure (e.g. Pearson correlation is the same for Fig. 1-6) to make these figures comparable and to draw further conclusions.

Author Response

Question 1:  I miss some discussion on how the numerical computation are carried out

Answer 1- Thank you very much for your advice. We have discussed in details on how the numerical computation are carried out in the revised manuscript. 

Question 2:  The authors discuss some plots trying different copula structures but they barely say a word on how the simulations are done.

Answer 2- Thank you very much for your advice. We have included more details to discuss all the plots of different copula structures in the revised manuscript. 

Question 3:  Some typos

Answer 3- Thank you very much for pointing out our typos.  We have revised our paper to take care of all typos, including the following

Reviewer 2 Report

In this paper, the authors derive general formulas to determine the density and distribution for the product of an arbitrary but finite number of random variables. These formulas rely on the use of copulas to account for the dependence structure between the random variables. As the formulas obtained are not explicit, the authors also discuss some numerical simulations to illustrate the utility of their approach. 

In my opinion, the results presented are interesting and worthy of being published. Although technically the computations are pretty straightforward, I feel that the value of the work is in the original approach to deal with products of random variables via copula techniques. I am not aware of similar work in the literature.

The results appear to be technically correct and well presented. However, I miss some discussion on how the numerical computations are carried on. The authors discuss some plots trying different copula structures but they barely say a word on how the simulations are done. Moreover, the paper would benefit from a careful reading from an English speaking native. There are plenty of small language mistakes and statements that do not sound natural.

Some other typos/remarks:

Page1: The first sentence in the abstract should be reformulated. To say that “the problem that you are interested in is one of the most important problems” is not informative. Maybe you should add in which contexts this problem is very important (risk management for instance).

Page 3: In equation (1), it should be w_1+w_2=1. On the other hand, shouldn't these weights be also positive?

Page 4: In definition 2, change “C has grounded” by “C is grounded”.

Page 5: In Theorem 1. (X1,X2) is a continuous vector or an absolutely continuous vector? Moreover, change “sgn()” by “sgn(·)” or “sgn”.

Author Response

Question 4a: The first sentence in the abstract should be reformulated. To say that “the problem that you are interested in is one of the most important problems” is not informative. Maybe you should add in which contexts this problem is very important (risk management for instance).

Answer 4a: Thank you very much for your advice. We have amended the sentence to be:

The problem of determining distributions of the functions of random variables is one of the most important problems in statistics and applied mathematics because distributions of functions have wide range of applications in numerous areas in economics, finance, risk management, science, and others.

Question 4b: In equation (1), it should be w_1+w_2=1.  On the other hand, shouldn’t these weights be also positive?

Answer 4b:  Thank you very much for pointing out our typo. We have corrected it in our revised manuscript.

We note that the result holds true for any w_1, w_2 in real set R\{0}. However, in the construction of portfolio, it must satisfy w_1+w_2=1, the weights can be negative in the situations of short selling.  We have included this information in our revised manuscript.

Question 4c: In definition 2, change “C has grounded” to “C is grounded”.

Answer 4c:  Thank you very much for pointing out our typo. We have corrected it in our revised manuscript.

Question 4d: In Theorem 1, (X_1,X_2) is a continuous vector or an absolutely continuous vector?

Answer 4d: Thank you very much for your helpful comments. Yes, we request both X_1 and X_2 to be absolutely continuous to guarantee the existence of density functions.

We have included the above information in our revised manuscript.

We hope that you will find this manuscript suitable to be included in an upcoming issue of your publication.

Author Response File: Author Response.docx

Reviewer 3 Report

The paper is of interest as it presents some useful results. The results are technically correct, as far as I can see, and given the usefulness of results, publication is warranted. Some improvement in the language is required. For example, in the abstract, the authors write,"We find that different types of copulas strongly affected behavior of distributions differently", whereas, it should have been, "We find that different types of copulas affect behavior of distributions differently". Little things like that are noticeable in the manuscript so a careful reading would help.

Author Response

Question；

 The paper would benefit from a careful reading from a English speaking native

Answer；

Thank you very much for your advice. We have polished our writing carefully.