# Talk:Cumulants

I have made a few minor edits directly to the text (removing repeated words or parentheses, adding missing parentheses, things of that nature). In addition I have several small queries to the authors, as follows:

1) In the section "Calculus of cumulants", in the fourth line, I'm not sure what "This freedom" refers to. Might there be a more clear way of introducing this idea? In the next two sentences, the word "formal" is used three times; the sentences would flow better without this repetition.

2) Is it worth adding more on the usefulness of umbrae and related ideas to statistical applications?

3) In the subsection "Edgeworth approximation" there appears to be a minor notational typo that I hesitated to correct myself because it has implications throughout the paragraphs that follow. It seems that both the MGF and the CGF are denoted by $K(\xi)$, instead of one being denoted $M(\xi)$.

4) Right before equation (9), should "... that the inversion process is properly smooth" read "... if the inversion process..." or "since the inversion process..." (or maybe it's enough to delete the word "that") ? The sentence as written doesn't parse but I wasn't entirely sure of the authors' intention.

5) Right after equation (9), what is S? This presumably stands for the sum of some number of i.i.d. copies of a random variable (as in the section "Properties"), but it's not clear, especially since Y is stated to arise from the sum of i.i.d. summands itself.

6) The section "Samples and sub-samples" seems rather disconnected from the rest of the article. Can the connections be made more explicit?

## Author Kolassa : Note to Revier A

Dear Reviewer A, Thanks for making these changes. I assume that you don't require any changes from us. Is this right? Thanks again, John Kolassa

## Author Kolassa : Note to Reviewer B

>I have made a few minor edits directly to the text (removing repeated words or >parentheses, adding missing parentheses, things of that nature). In addition I >have several small queries to the authors, as follows:

Many thanks. >1) In the section "Calculus of cumulants", in the fourth line, I'm not sure what >"This freedom" refers to. Might there be a more clear way of introducing this >idea? In the next two sentences, the word "formal" is used three times; the >sentences would flow better without this repetition.

Wordsmithing "freedom" is still in process. The second and third occurrence of "formal" have been removed.

>2) Is it worth adding more on the usefulness of umbrae and related ideas to >statistical applications?

Absolutely. This is in progress.

>3) In the subsection "Edgeworth approximation" there appears to be a minor >notational typo that I hesitated to correct myself because it has implications >throughout the paragraphs that follow. It seems that both the MGF and the CGF are >denoted by $K(\xi)$, instead of one being denoted $M(\xi)$.

Two occurrences of K have been replaced by M.

>4) Right before equation (9), should "... that the inversion process is properly >smooth" read "... if the inversion process..." or "since the inversion process..." >(or maybe it's enough to delete the word "that") ? The sentence as written doesn't >parse but I wasn't entirely sure of the authors' intention.

Yow! Did I really write that sentence? It's simplified by being divided into two sentences, and changing "that" to "since".

>5) Right after equation (9), what is S? This presumably stands for the sum of some >number of i.i.d. copies of a random variable (as in the section "Properties"), but >it's not clear, especially since Y is stated to arise from the sum of i.i.d. >summands itself.

S is replaced by Y.

>6) The section "Samples and sub-samples" seems rather disconnected from the rest >of the article. Can the connections be made more explicit?

We've added a sentence noting that this property keeps cumulant estimators that are calculated from a subsample but unbiased for the sample unbiased for the population as well.

Thanks again, John Kolassa

## Author Kolassa : Addendum to Referee B

Sorry about the e-mail alert. When I hit the button, I thought that I'd have the opportunity to say that not all changes were made, but apparently the software doesn't allow that. We're still in progress on two changes requested. John Kolassa

## Reviewer B:

Thanks for clarifying the points in my discussion. I have nothing more to add.

## Reviewer C:

I have only minor comments as follows:

1. In paragraph of "Definition under less restrictive conditions" on page 4, ":" in the 3rd sentence should be ";".

2. Under "Edgeworth approximation" section, "Exponentiating" in the 2nd paragraph should be "Expanding"