Talk:Unimodality

Split proposal

Apparently, the two pages unimodal function and unimodal distribution were merged in 2010 for some reason. I think it would make more sense to change Unimodality to a disambiguation page with links to these other pages, since the two concepts are quite different. Isheden (talk) 11:16, 6 May 2012 (UTC)

I don't see what we gain by splitting to two small closely related subjects. Having them together enables the reader to get a wider perspective of the subject. --Muhandes (talk) 13:16, 7 May 2012 (UTC)

The question is how closely related the two subjects really are. The page mode (statistics) does not mention any functions other than pdf and cdf of a distribution, so it is unclear what the "mode" of a function would be. Are there any sources supporting that the concept of a mode in statistics can be applied to functions also? I'm asking because this is dubious in light of "As the term "modal" applies to data sets and probability distribution, and not in general to functions, the definitions above do not apply." Isheden (talk) 14:47, 7 May 2012 (UTC)

Moreover, the lead sentence "Unimodality is a term used in several contexts in mathematics." would be a typical way to begin a disambiguation page. Isheden (talk) 14:49, 7 May 2012 (UTC)

That the term is in use is unquestionable. The article provides [1] as source, but I easily found it used in many articles e.g. [2] [3]. I agree that the mode cannot easily be applied to functions, but my conclusion from this is the opposite of yours. If we had unimodal function on itself it would be unclear since the term "mode of function" is unclear. However, when put together with distribution functions, the connection becomes clearer. I see unimodal functions as an extension of unimodal distributions, and as such find it hard to understand on its own. --Muhandes (talk) 15:33, 7 May 2012 (UTC)

Yes, it is clear that unimodality, unimodal function, and unimodal distribution are all commonly used. However, I'd say that unimodal function is a concept in mathematical optimization that has borrowed the terminology from the concept of a unimodal distribution in statistics. In the literature, "unimodality" seems to refer to either one of these concepts, depending on the field, but not to both. If you search for either "unimodal function" or "unimodal distribution" on Google books, you get thousands of hits. However, if you search for both of them together, there are very few hits. Isheden (talk) 16:03, 7 May 2012 (UTC)

I now see where you are heading. I'm not sure though that the term "unimodal function" is restricted to the area of optimization. This is especially true for the matters discussed under "Other extensions". For example, S-unimodality is mentioned in articles about statistics (see e.g. [4]). in other words, if indeed the two terms were used in two fields, it may have made sense to split. But it appears like unimodal functions may be a cross-field subject. --Muhandes (talk) 11:11, 8 May 2012 (UTC)

Incorrect Definition of Unimodality

The text proposes as "strict definition" of unimodal as a distribution that has one "single" maxima, then proposes others "less strict". This is wrong, not in sync with the literature, and the article needs to be rewritten. Bimodal would then only hold if two modes are exactly equal, near impossible. The other "less strict" definitions are the ones that are in common use. I will need to remove all the discussions related to this "strict" definition. Limit-theorem (talk) 19:44, 13 October 2013 (UTC)

The incorrect definition remained but I have now tried to remove it. I hope I got it right, but I am sure I made the article better. MathHisSci (talk) 17:29, 5 June 2018 (UTC)

Copyright problem removed

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Skewness and kurtosis inequality

Two bounds are given for skewness^2 - kurtosis:

$\gamma ^{2}-\kappa \leq {\frac {5}{6}}$

$\gamma ^{2}-\kappa \leq {\frac {186}{125}}$

The second bound is both redundant and more complex. Shouldn't it be removed? Or is there some subtlety to justify it which needs to be pointed out?

• The previous 5/6 fraction was corrected to 6/5 - from the Rohatgi Szekely article. It is not redundant with the 186/125 fraction. In some cases the latter will yield a positive for unimodality where the former (6/5) will not.--Flavonoid (talk) 23:25, 20 May 2014 (UTC)

• However, Bhattacharyya claims $\kappa -\gamma ^{2}-1\geq 0$ due to "...the positive semidefiniteness of moment matrices". I cannot verify this claim:

Communications in Statistics - Theory and Methods

Publication details, including instructions for authors and subscription information:

http://www.tandfonline.com/loi/lsta20

One sided chebyshev inequality when the first four moments are known

B.B. Bhattacharyya a

a North Carolina State University, Raleigh, North Carolina, 27695, U.S.A

Version of record first published: 27 Jun 2007. — Preceding unsigned comment added by 203.110.235.14 (talk) 03:03, 6 March 2015 (UTC)

That inequality uses plain kurtosis while the other ones use excess kurtosis. In terms of excess kurtosis, your inequality is $\gamma ^{2}-\kappa \leq 2$ which is weaker than both previous bounds. --Unimodal aunt (talk) 16:41, 3 April 2021 (UTC)

• The first bound is incorrect in the general setting, as shown by the counterexample I just added to the page. The authors of the second paper politely call it a "conjecture", but the first paper did present it as a theorem. Their proof was wrong because the mixture of two distributions satisfying the bound does not always satisfy the bound. This page has contained the incorrect result since 2012! --Unimodal aunt (talk) 15:40, 3 April 2021 (UTC)

Saying "incorrectly claimed" is WP:OR. You may need to tone it down by saying "under the assumption that the mode..." Limit-theorem (talk) 16:00, 3 April 2021 (UTC)

But the 1989 paper does not mention those assumptions, so the toned-down version would not be supported by the source either unless we silently remove all references to that paper. I don't think mentioning that the first paper was disproved fails WP:OR, and specifically SYNTH since this is a conclusion that can only be ascertained by consulting both papers. See the "SYNTH is not unpublishably unoriginal" test in WP:SYNTHNOT:

When you look at a case of putative SYNTH, apply the following test. Suppose you took this claim to a journal that does publish original research. Would they (A) vet your article for correctness, documentation, and style, and publish it if it met their standards in those areas? Or would they (B) laugh in your face because your "original research" is utterly devoid of both originality and research, having been common knowledge in the field since ten years before you were born? If you chose (B), it's not original research -- even if it violates the letter of WP:SYNTH.

It is absolutely clear that any professional mathematician reading the two papers would come to the conclusion that the second paper is saying very diplomatically that the bound given in the first paper is incorrect in the general case. No one could get an article published that disproves the first paper, regardless of how explicit they were in their wording, because this is precisely what the second paper did except for the understated wording.

And unfortunately "understated" can also mean "unclear" for the average reader. Both papers have been cited by this page for the last 8.5 years, yet no one noticed that the second paper contradicts the first. Let's call a spade a spade! --Unimodal aunt (talk) 16:41, 3 April 2021 (UTC)