On Underdispersed Count Kernels for Smoothing Probability Mass Functions

Round 1

Reviewer 1 Report

 Two novel underdispersed count kernels, specifically the double Poisson and gamma-count ones are proposed in the study. They have been developed by using the proposed mean-dispersion method. Also, the authors have considered the integrated squared error method to select as quickly and efficiently as possible the bandwidth of their corresponding estimations. Through simulation experiments and real count data analysis, it is demonstrated that these kernels perform better than the binomial kernel, while falling between the CoM-Poisson kernel smoothing (which performs the best) and the binomial kernel (which performs the worst).  The study is interesting and the following issues should be dealt with before publication. First, some application of the underdispersed count kernels for smoothing probability mass functions can be introduced. Second, some insightful explanation can be given after each theorem. Third, the whole paper should be thoroughly checked before resubmition.

Author Response

Many thanks for all these positive comments. We recall some applications of some underdispersed count kernels in the sentence "The reader can also refer to [12-14] for some applications of discrete kernels in survey sampling, model specification test, discriminant analysis, respectively and more generally to Li and Racine [15] ". Please also notice that all definitions and remarks have been motivated by examples. Finally, we carefully check all this revised version.

Reviewer 2 Report

This paper considered ‘count kernel for smoothing probability mass functions’. It is good writing. If it possible, could the authors present the time consuming of this method?

Author Response

 Please refer to Table 1 and the related comments.

Reviewer 3 Report

Dear Authors,

You provide an excellent paper, fast without errors or typos. I didn't verify all your formulae, but they seem correct. I spotted only a few ambiguities (see below). I didn't understand what was new in the first appendix (Was it material already published elsewhere?). Perhaps you have to explain more about the utility of this appendix. 

With best regards

l 7: "these kernels numerically work very well". What does that mean?

l 10: "the overall efficiency" What is "efficiency" in this context?

l 33: "their estimators do not converge" Which estimators? Estimators of what?

l 96: "we normalize by T=1" Why? And is it in (4)?

l 103: "this mean can be logarithmic or approximatively linear" Why?

l 107: "the variance ... is proportional to..." It is not so evident to understand why.

l 130: "If a solution to this..." How can we obtain a solution or verify if a solution exists". Furthermore, your notation is not really used afterwards.

l 150: "at the origin x=0 and inside x=5" What is "inside x=5"?

l 178: "...kernel estimators are efficient" What is efficient here?

See above

Author Response

Dear Authors,

You provide an excellent paper, fast without errors or typos. I didn't verify all your formulae, but they seem correct. I spotted only a few ambiguities (see below). I didn't understand what was new in the first appendix (Was it material already published elsewhere? Not in the purpose of count kernels). Perhaps you have to explain more about the utility of this appendix. 

With best regards

Authors response: We have completed the title of this first appendix and this material is new in the literature (not yet published).

- l 7: ``these kernels numerically work very well''.   What does that mean?\\
Authors response: The sentence is now "Despite a challenging problem for obtaining explicit expressions, these kernels, numerically, properly smooth densities."
 
-l 10: ``the overall efficiency'' What is ``efficiency'' in this context?
Authors response: Efficiency is related to accuracy here. Thus, the sentence is now "Thus, the overall accuracy of two newly suggested kernels appears to be between the two old ones."

- l 33: ``their estimators do not converge'' Which estimators? Estimators of what?
Authors response: Thank you. We mean here ``corresponding estimators''. The sentence becomes ``... their corresponding estimators do not converge.''

- l 96: ``we normalize by $T=1$'' Why? And is it in (4)?
We here consider $T=1$ ?
Authors response: The sentence is rewritten as follows ``...$T$ can be set to one, without loss of generality.''

- l 103: ``this mean can be logarithmic or approximatively linear'' Why?
Authors response: At first sight,  "precisely, by zooming in, we notice that the shape of the curve is logarithmic or approximately linear."

- l 107: ``the variance ... is proportional to...'' It is not so evident to understand why.
Authors response: Now the sentence is ``Hence, the variance of the gamma-count distribution can be seen as a function of $\alpha>0$.''