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The formula
might be written
The constant term is found by setting t = 0:
Zero is not a cumulant, and so the function
better deserves the name 'cumulant generation function'.
Bo Jacoby 12:55, 4 January 2006 (UTC)
I understand your reservations against changing conventions. Note, however, the tempting simplification obtained by differentiation.
Bo Jacoby 12:21, 31 January 2006 (UTC)
What do you mean? Please explain first and conclude later. The two definitions allow the same operations. The new definition just does not contain a superfluous zero constant term. The graph of the new cumulant-generating function passes through (0, μ) having the slope σ2. Curvature shows departure from normality: μ+σ2t. Bo Jacoby 09:14, 1 February 2006 (UTC)
I'll comment further on Bo Jacoby's comments some day. But for now, let's note that what is in the article has been the standard convention in books and articles for more than half a century, and Wikipedia is not the place to introduce novel ideas. Michael Hardy 22:55, 4 April 2006 (UTC)
I came randomly to see the article : no explanation about a cumulant were in view. A TOC was followed by formulas.
We math people love what we do. Let us try to do more : explain what we do (for this, I need help).
P.S. Wolfram, for example, gives links to : Characteristic Function, Cumulant-Generating Function, Fourier Transform, k-Statistic, Kurtosis, Mean, Moment, Sheppard's Correction, Skewness, Unbiased Estimator, Variance. [Pages Linking Here]. Though I cannot tell if it is pertinent here, maybe a little check might be done ? Thanks again. -- DLL 18:56, 9 June 2006 (UTC)
"Innerproduct", I see that you commented nearly four years after the comment you're replying to. It should be perfectly obvious that that comment was about the article as it existed in 2006, and is no longer relevant to the article in its present form.
Your proposed general rule is very bad (even though in some particular cases it makes sense); following it extensively would require people to clean up after you. Michael Hardy ( talk) 22:36, 2 April 2010 (UTC)
(please forgive my english)
I think that in the formula
the number |B| of elements in B should be replaced by the number of blocks in .
For example, in the given case n=3
the constant before the term E(XYZ) (which corresponds to : only one block of 3 items, i.e. |\pi|=1 and \pi={B} with |B|=3) is and not .
We need an intro -- dudzcom 04:52, 24 December 2006 (UTC)
I think we should have a section on k-statistics. Could someone knowledgeable write a section describing them and explaining why they are unbiased estimators for the cumulants. Ossi 18:04, 30 December 2006 (UTC)
I have been trying to find information about unbiased and other estimators for ratios of powers of cummulants. In particular, I am interested in estimators for the particular ratio . I can use k-statistics to estimate this ratio as simply where is the k-statistic. This should work, but it is a biased estimator. Are there better, unbiased estimators? User:155.101.22.76 (Talk) 28 Oct 2008
In my case , and this reduces to:
Which is precisely the same estimator I derived using other methods also assuming a normal distribution. I did not know that you could reduce bias using jackknifing and bootstrapping methods. Doh! That is great. The distributions I am working with are close to normal. I should be able to use the above MVUE and then reduce any remaining bias using jackknifing or bootstrapping. Thanks. -- Stanthomas ( talk) 17:34, 30 October 2008 (UTC)
It appears that you can reconstruct a function from its cumulants; that is, it seems like the cumulants define a "basis" of sorts the same way the sin and cos functions define a Fourier basis. Of course, a function isn't a linear combination of its cumulants, so it's not a linear basis, but in some sense it still seems like a basis. Comments? 155.212.242.34 ( talk) 22:23, 11 December 2007 (UTC)
Quote: Some writers prefer to define the cumulant generating function, via the characteristic function, as h(t) where
I suppose the formula should be:
Is there a reference? Bo Jacoby ( talk) 00:43, 20 March 2008 (UTC).
At the end of the intro, the final sentance says: "This characterization of cumulants is valid even for distributions whose higher moments do not exist." This seems to dangle somewhat...
Melcombe ( talk) 09:30, 17 April 2008 (UTC)
I want to know more about Joint Cumulants, but this section made no reference to any books or papers. Any suggestions? Thanks! Yongtwang ( talk) 13:47, 13 May 2010 (UTC)
The article states that the cumulant-generating function is always convex (not too hard to prove). I wonder if the converse holds: any convex function (+ maybe some regularity conditions) can be a cumulant-generating function of some random variable. // stpasha » 20:03, 2 March 2011 (UTC)
I was reading this article to get a more broad background on the cumulant expansion, which is useful in quantum mechanical simulations of spectroscopic signals (absorption, pump-probe, raman, etc). I was somewhat surprised not to see quantum mechanics mentioned at all in the article. The source that I'm currently following on this topic:
Shaul Mukamel's "Principles of Nonlinear Optical Spectroscopy" (ISBN: 0-19-513291-2).
The expansions are debuted in Ch2, "Magnus Expansion". Ch 8 is also devoted entirely to their practical use.
Side note: It amused me that there were "citation needed" marks on the phrase, "Note that expectation values are sometimes denoted by angle brackets". This notation is so ubiquitous in quantum mechanics that one could literally pick up any quantum textbook and insert it as a "source" to verify that this is common practice. Certainly the book I just mentioned could count as such a source. —Preceding unsigned comment added by 24.11.171.13 ( talk) 23:26, 20 May 2011 (UTC)
I looked in the comments specifically to discuss the "citation needed" marks for angle bracket denotation of expectation values. It's like asking for a citation that addition is sometimes denoted with a plus sign. I'm going to remove it and someone can put it back in if they feel it's really necessary. Gregarobinson ( talk) 16:14, 17 June 2011 (UTC)
Some formulas in section "Relation to statistical physics" are wrong. Instead of:
the formula should read:
as found in any relevant textbook or the wiki page for the partition function itself. This breaks the following argument linking F(\beta) to the cumulant generating function for the energy \log Z, as Z is no longer an average. The same critic holds for the grand potential at the end of the section, which is also a sum, not an average.
The equations linking E and C to the corresponding cumulants of the energy are still valid, since the cumulants equal the moments (section "Some properties of cumulants"). However, the interpretation in terms of moments is quite widespread, and in fact the equation:
is considered a postulate, in which the energy is linked to an average. The addition of usage of cumulants in stat. mech. that can't be expressed more naturally in terms of moments should be made, if such an usage exists.
Futhermore, the section doesn't cite any source, and none of the article's sources seems relevant at first sight.
These three points make me feel the whole section is rather weak. I suggest it should be deleted. -- Palatosa ( talk) 19:53, 19 April 2013 (UTC)
I very much liked the section, except that was very disappointed that it did not actually included the definition of the CGF for this system. Chris2crawford ( talk) 11:44, 17 July 2020 (UTC)
There is something strange with the statement "The cumulant-generating function will have vertical asymptote(s) at the infimum of such c, if such an infimum exists etc". Note that x is a negative number here. Something being O(exp(2x)) is a tougher requirement than being O(exp(x)) when x tends to minus infinity. You get the toughest requirement possible by finding the supremeum over c. But changing infimum to supremum doesnt seem right either. Should there be some sign change also? — Preceding unsigned comment added by 89.236.1.222 ( talk) 22:12, 17 October 2014 (UTC)
The definition given requires the moment generating function to exist. Rather than change the definition to use the characteristic function, we just need a note that the relation between moments and cumulants given later, can be used as the definition. TerryM--re ( talk) 04:18, 12 February 2015 (UTC)
This article is very good and informative but someone has a flag on it to improve citations. Standard approaches do not apply to mathematical subjects, where inline citations are not as frequent, and usually one cites a theorem or a result, with plenty of references in back. Limit-theorem ( talk)
The article on moments includes a table mapping named properties of distributions (mean, variance, skewness, etc.) to the various kinds of moments and cumulants. This table suggests there is a distinction between "raw" and "standardized" cumulants, though I can find no specific explanation of how these two concepts differ. I gather that raw cumulants are the ordinary cumulants discussed in this article, while standardized cumulants are those computed for distributions normalized to have zero mean and unit variance. Accordingly, the first two standardized cumulants are 0 and 1, but all following standardized cumulants appear to be something like for i > 2 (extrapolated from the formulae for skewness and kurtosis when expressed in terms of raw cumulants). Rriegs ( talk) 21:58, 1 April 2016 (UTC)
As an aside, the value (i.e. the mean squared divided by the variance) discussed elsewhere in this talk page is just the above formula squared for i = 1, though this is not a standardized cumulant. Is there a name or special significance for this value? Rriegs ( talk) 21:58, 1 April 2016 (UTC)
Can anyone point to a reference which proves the multi-linearity property for joint cumulants? Or if the possible give a short argument? I find it very intriguing. Manoguru ( talk) 06:21, 25 August 2017 (UTC)
@ Michael Hardy: thank you for your recent edits. However, they seem to duplicate sections already present later in the article. I leave it to you how best to condense things so that there isn't that duplication. For example the discussion about the relationship of cumulants to central moments is later done in terms of κ and μ relationships. — Quantling ( talk | contribs) 02:48, 9 November 2021 (UTC)
The section Cumulants_of_some_discrete_probability_distributions contains the wrong formula
Replacing it with
Bo Jacoby ( talk) 16:38, 10 November 2021 (UTC)
Currently it starts by mentioning a (joint) cumulant generating function , and then it jumps to: A consequence is that ...
I don't think, this is very clear. At least I don't understand it. My confusion starts from the fact that the cumulants forms a sequence of numbers which are the coefficients of a univariate power series. Here is multivariate. So I would expect not one single joint cumulant but a collection of joint cumulants .
If someone could clarify this, it would be fantastic.
Also a (book/article) reference for the joint cumulants would be very welcome, as in many books only the (simple) cumulants are treated. Bongilles ( talk) 08:42, 24 November 2023 (UTC)
I have done further changes. This should be more clear and consistant now. I also added references. Bongilles ( talk) 10:09, 30 November 2023 (UTC)
This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||||||||||||
|
The formula
might be written
The constant term is found by setting t = 0:
Zero is not a cumulant, and so the function
better deserves the name 'cumulant generation function'.
Bo Jacoby 12:55, 4 January 2006 (UTC)
I understand your reservations against changing conventions. Note, however, the tempting simplification obtained by differentiation.
Bo Jacoby 12:21, 31 January 2006 (UTC)
What do you mean? Please explain first and conclude later. The two definitions allow the same operations. The new definition just does not contain a superfluous zero constant term. The graph of the new cumulant-generating function passes through (0, μ) having the slope σ2. Curvature shows departure from normality: μ+σ2t. Bo Jacoby 09:14, 1 February 2006 (UTC)
I'll comment further on Bo Jacoby's comments some day. But for now, let's note that what is in the article has been the standard convention in books and articles for more than half a century, and Wikipedia is not the place to introduce novel ideas. Michael Hardy 22:55, 4 April 2006 (UTC)
I came randomly to see the article : no explanation about a cumulant were in view. A TOC was followed by formulas.
We math people love what we do. Let us try to do more : explain what we do (for this, I need help).
P.S. Wolfram, for example, gives links to : Characteristic Function, Cumulant-Generating Function, Fourier Transform, k-Statistic, Kurtosis, Mean, Moment, Sheppard's Correction, Skewness, Unbiased Estimator, Variance. [Pages Linking Here]. Though I cannot tell if it is pertinent here, maybe a little check might be done ? Thanks again. -- DLL 18:56, 9 June 2006 (UTC)
"Innerproduct", I see that you commented nearly four years after the comment you're replying to. It should be perfectly obvious that that comment was about the article as it existed in 2006, and is no longer relevant to the article in its present form.
Your proposed general rule is very bad (even though in some particular cases it makes sense); following it extensively would require people to clean up after you. Michael Hardy ( talk) 22:36, 2 April 2010 (UTC)
(please forgive my english)
I think that in the formula
the number |B| of elements in B should be replaced by the number of blocks in .
For example, in the given case n=3
the constant before the term E(XYZ) (which corresponds to : only one block of 3 items, i.e. |\pi|=1 and \pi={B} with |B|=3) is and not .
We need an intro -- dudzcom 04:52, 24 December 2006 (UTC)
I think we should have a section on k-statistics. Could someone knowledgeable write a section describing them and explaining why they are unbiased estimators for the cumulants. Ossi 18:04, 30 December 2006 (UTC)
I have been trying to find information about unbiased and other estimators for ratios of powers of cummulants. In particular, I am interested in estimators for the particular ratio . I can use k-statistics to estimate this ratio as simply where is the k-statistic. This should work, but it is a biased estimator. Are there better, unbiased estimators? User:155.101.22.76 (Talk) 28 Oct 2008
In my case , and this reduces to:
Which is precisely the same estimator I derived using other methods also assuming a normal distribution. I did not know that you could reduce bias using jackknifing and bootstrapping methods. Doh! That is great. The distributions I am working with are close to normal. I should be able to use the above MVUE and then reduce any remaining bias using jackknifing or bootstrapping. Thanks. -- Stanthomas ( talk) 17:34, 30 October 2008 (UTC)
It appears that you can reconstruct a function from its cumulants; that is, it seems like the cumulants define a "basis" of sorts the same way the sin and cos functions define a Fourier basis. Of course, a function isn't a linear combination of its cumulants, so it's not a linear basis, but in some sense it still seems like a basis. Comments? 155.212.242.34 ( talk) 22:23, 11 December 2007 (UTC)
Quote: Some writers prefer to define the cumulant generating function, via the characteristic function, as h(t) where
I suppose the formula should be:
Is there a reference? Bo Jacoby ( talk) 00:43, 20 March 2008 (UTC).
At the end of the intro, the final sentance says: "This characterization of cumulants is valid even for distributions whose higher moments do not exist." This seems to dangle somewhat...
Melcombe ( talk) 09:30, 17 April 2008 (UTC)
I want to know more about Joint Cumulants, but this section made no reference to any books or papers. Any suggestions? Thanks! Yongtwang ( talk) 13:47, 13 May 2010 (UTC)
The article states that the cumulant-generating function is always convex (not too hard to prove). I wonder if the converse holds: any convex function (+ maybe some regularity conditions) can be a cumulant-generating function of some random variable. // stpasha » 20:03, 2 March 2011 (UTC)
I was reading this article to get a more broad background on the cumulant expansion, which is useful in quantum mechanical simulations of spectroscopic signals (absorption, pump-probe, raman, etc). I was somewhat surprised not to see quantum mechanics mentioned at all in the article. The source that I'm currently following on this topic:
Shaul Mukamel's "Principles of Nonlinear Optical Spectroscopy" (ISBN: 0-19-513291-2).
The expansions are debuted in Ch2, "Magnus Expansion". Ch 8 is also devoted entirely to their practical use.
Side note: It amused me that there were "citation needed" marks on the phrase, "Note that expectation values are sometimes denoted by angle brackets". This notation is so ubiquitous in quantum mechanics that one could literally pick up any quantum textbook and insert it as a "source" to verify that this is common practice. Certainly the book I just mentioned could count as such a source. —Preceding unsigned comment added by 24.11.171.13 ( talk) 23:26, 20 May 2011 (UTC)
I looked in the comments specifically to discuss the "citation needed" marks for angle bracket denotation of expectation values. It's like asking for a citation that addition is sometimes denoted with a plus sign. I'm going to remove it and someone can put it back in if they feel it's really necessary. Gregarobinson ( talk) 16:14, 17 June 2011 (UTC)
Some formulas in section "Relation to statistical physics" are wrong. Instead of:
the formula should read:
as found in any relevant textbook or the wiki page for the partition function itself. This breaks the following argument linking F(\beta) to the cumulant generating function for the energy \log Z, as Z is no longer an average. The same critic holds for the grand potential at the end of the section, which is also a sum, not an average.
The equations linking E and C to the corresponding cumulants of the energy are still valid, since the cumulants equal the moments (section "Some properties of cumulants"). However, the interpretation in terms of moments is quite widespread, and in fact the equation:
is considered a postulate, in which the energy is linked to an average. The addition of usage of cumulants in stat. mech. that can't be expressed more naturally in terms of moments should be made, if such an usage exists.
Futhermore, the section doesn't cite any source, and none of the article's sources seems relevant at first sight.
These three points make me feel the whole section is rather weak. I suggest it should be deleted. -- Palatosa ( talk) 19:53, 19 April 2013 (UTC)
I very much liked the section, except that was very disappointed that it did not actually included the definition of the CGF for this system. Chris2crawford ( talk) 11:44, 17 July 2020 (UTC)
There is something strange with the statement "The cumulant-generating function will have vertical asymptote(s) at the infimum of such c, if such an infimum exists etc". Note that x is a negative number here. Something being O(exp(2x)) is a tougher requirement than being O(exp(x)) when x tends to minus infinity. You get the toughest requirement possible by finding the supremeum over c. But changing infimum to supremum doesnt seem right either. Should there be some sign change also? — Preceding unsigned comment added by 89.236.1.222 ( talk) 22:12, 17 October 2014 (UTC)
The definition given requires the moment generating function to exist. Rather than change the definition to use the characteristic function, we just need a note that the relation between moments and cumulants given later, can be used as the definition. TerryM--re ( talk) 04:18, 12 February 2015 (UTC)
This article is very good and informative but someone has a flag on it to improve citations. Standard approaches do not apply to mathematical subjects, where inline citations are not as frequent, and usually one cites a theorem or a result, with plenty of references in back. Limit-theorem ( talk)
The article on moments includes a table mapping named properties of distributions (mean, variance, skewness, etc.) to the various kinds of moments and cumulants. This table suggests there is a distinction between "raw" and "standardized" cumulants, though I can find no specific explanation of how these two concepts differ. I gather that raw cumulants are the ordinary cumulants discussed in this article, while standardized cumulants are those computed for distributions normalized to have zero mean and unit variance. Accordingly, the first two standardized cumulants are 0 and 1, but all following standardized cumulants appear to be something like for i > 2 (extrapolated from the formulae for skewness and kurtosis when expressed in terms of raw cumulants). Rriegs ( talk) 21:58, 1 April 2016 (UTC)
As an aside, the value (i.e. the mean squared divided by the variance) discussed elsewhere in this talk page is just the above formula squared for i = 1, though this is not a standardized cumulant. Is there a name or special significance for this value? Rriegs ( talk) 21:58, 1 April 2016 (UTC)
Can anyone point to a reference which proves the multi-linearity property for joint cumulants? Or if the possible give a short argument? I find it very intriguing. Manoguru ( talk) 06:21, 25 August 2017 (UTC)
@ Michael Hardy: thank you for your recent edits. However, they seem to duplicate sections already present later in the article. I leave it to you how best to condense things so that there isn't that duplication. For example the discussion about the relationship of cumulants to central moments is later done in terms of κ and μ relationships. — Quantling ( talk | contribs) 02:48, 9 November 2021 (UTC)
The section Cumulants_of_some_discrete_probability_distributions contains the wrong formula
Replacing it with
Bo Jacoby ( talk) 16:38, 10 November 2021 (UTC)
Currently it starts by mentioning a (joint) cumulant generating function , and then it jumps to: A consequence is that ...
I don't think, this is very clear. At least I don't understand it. My confusion starts from the fact that the cumulants forms a sequence of numbers which are the coefficients of a univariate power series. Here is multivariate. So I would expect not one single joint cumulant but a collection of joint cumulants .
If someone could clarify this, it would be fantastic.
Also a (book/article) reference for the joint cumulants would be very welcome, as in many books only the (simple) cumulants are treated. Bongilles ( talk) 08:42, 24 November 2023 (UTC)
I have done further changes. This should be more clear and consistant now. I also added references. Bongilles ( talk) 10:09, 30 November 2023 (UTC)