Smooth Versions of the Mann–Whitney–Wilcoxon Statistics
John E. Kolassa
Round 1
Reviewer 1 Report
This manuscript looks entirely reasonable. I didn't check all of the math. My biggest concern is around the central limit theorem argument. Otherwise, here are my comments:
1. Conditions on W_{m,n}: I don't know what is meant by W_{m,n(u)->I_{(-\infty,\infty)}(u): I don't know what the I function is, and I don't know what kind of convergence is implied.
2. Proof of (i): The same definitional issues as in point 1 above apply here to I_{(0,\infty)}(x); furthermore, I thing tha tthe arguemnts of the in-line stat
ement representing the first line of page 3 should have the same argument to W and to I.
The authors quite correctly use pointwise convergence of the characteristic functions to imply weak convergence, They then use this to imply convergence in mea
n. They make use of the Dominated Convergence Theorem, but I don't see the information needed to set up the dominating statement.
3. Proof of (ii): In the line below (10), W^2_{m,n} has an extra ).
4. Proof of (iii): I don't see how the last line of page 3 is obtained. There ought to be some "almost surely" / "with probability 1" arugment here, and I don't see it.
5. I don't see the argument for (iv) at all. B is the difference of two sample averages only if W_{m,n} has a very simple dependence on m and n, and in in the absense of independence, the difference between two asymptotically normal random variables is not necessarily asymptotically normal.
There also appear to be extra characters following some of the line numbers; see (23/3.7) on page 7, and some other places. Does it refer to section.page? I don't think it's necessary.
Author Response
Dear First Reviewer
June 6th, 2020
We thank your generous comments on the manuscript and have edited the manuscript to address your concerns.
Below is our responses to each point raised by you. We hope that we satisfyingly addressed them and that the manuscript will be now suitable for publication in Axioms.
Sincerely,
On behalf of all authors.
Netti Herawati, Ph.D.
Point 1. Conditions on W_{m,n}: I don't know what is meant by W_{m,n(u)->I_{(-\infty,\infty)}(u): I don't know what the I function is, and I don't know what kind of convergence is implied.
Response 1. Is a rectangular array of known distribution functions such that as m.
Point 2. Proof of (i): The same definitional issues as in point 1 above apply here to I_{(0,\infty)}(x); furthermore, I thing that the arguemnts of the in-line statement representing the first line of page 3 should have the same argument to W and to I. The authors quite correctly use pointwise convergence of the characteristic functions to imply weak convergence, They then use this to imply convergence in mean. They make use of the Dominated Convergence Theorem, but I don't see the information needed to set up the dominating statement.
Response 2. is the characteristic function of
Point 3. Proof of (ii): In the line below (10), W^2_{m,n} has an extra ).
Response 3. . already revised
Point 4. Proof of (iii): I don't see how the last line of page 3 is obtained. There ought to be some "almost surely" / "with probability 1" arugment here, and I don't see it.
Response 4. Proof of (iii) is based on showing E and then as m.
Point 5. I don't see the argument for (iv) at all. B is the difference of two sample averages only if W_{m,n} has a very simple dependence on m and n, and in in the absense of independence, the difference between two asymptotically normal random variables is not necessarily asymptotically normal.
Response 5. It is clearly stated
Point 6. There also appear to be extra characters following some of the line numbers; see (23/3.7) on page 7, and some other places. Does it refer to section.page? I don't think it's necessary.
Response 6. Has been revised
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Dear Authors,
I am glad I have this opportunity to review your manuscript entitled "A Smooth Version of the Mann-Whitney-Wilcoxon Statistics". Based on its title and abstract I was expecting a very interesting manuscript, but... see my comments/notes as follows (I will write it according to the structure of your manuscript):
Title
- it is too general, it should be specified
Abstract
- it is too general and short, the first part is ok, but after that describe your methodology and the most important, much more focus on your own results
Introduction
- clearly identify the aim of your study/research (not only explain the MWW procedure and your idea)
- line 41 "A huge number of studies have been carried out to investigate the properties..." - be more specific, name these studies and a little bit describe them (why it is important to mention them? what are the results of these studies?)
Large Sample Theory of...
- no comments, described step by step
Robustness of...
- no comments, described step by step
Other comments
- the structure of the manuscript should be modified (check the journal requirements again and incorporate each of them)
- add section "Discussion" and discuss/compare your results with already published studies (show us, what is your added value)
- add section "Conclusion" and summarize your study, identify its limits and possible further research (how these results could be used for further research?)
- extend the literature used, add more actual sources and cite current research published (actually you cited two pieces of literature after the year 2015 ), is the topic still actual? If yes, you should be able to find other appropriate literature.
Author Response
Dear Second Reviewer
June 6th, 2020
We thank your generous comments on the manuscript and have edited the manuscript to address your concerns.
All points raised by you have been answered in the text.
We hope that we satisfyingly addressed them and that the manuscript will be now suitable for publication in Axioms.
Sincerely,
On behalf of all authors.
Netti Herawati, Ph.D.
Poin 1. Title- it is too general, it should be specified
Response1: Has been revised
Point 2. Abstract. - it is too general and short, the first part is ok, but after that describe your methodology and the most important, much more focus on your own results
Response 2: Has been revised
Point 3. Introduction. clearly identify the aim of your study/research (not only explain the MWW procedure and your idea). - line 41 "A huge number of studies have been carried out to investigate the properties..." - be more specific, name these studies and a little bit describe them (why it is important to mention them? what are the results of these studies?)
Response 3: Has been revised
Point 4. Other comments. the structure of the manuscript should be modified (check the journal requirements again and incorporate each of them). - add section "Discussion" and discuss/compare your results with already published studies (show us, what is your added value). add section "Conclusion" and summarize your study, identify its limits and possible further research (how these results could be used for further research?)
Response 4: Discussion has been added according to the requirements, however the conclusion is not a mandatory as stated in the template structure for Axioms
Point 5. extend the literature used, add more actual sources and cite current research published (actually you cited two pieces of literature after the year 2015 ), is the topic still actual? If yes, you should be able to find other appropriate literature.
Response 5. The most recent references in our articles are 2019 and 2020. We consider them sufficient because we did not find any other recent literature related to our topic.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
I have checked the author's reply. Each of my comments has been discussed, so I do not have other ones.
