# Talk:Unimodality

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## 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)

## Skewness and kurtosis inequality

Two bounds are given for skewness^2 - kurtosis:

${\displaystyle \gamma ^{2}-\kappa \leq {\frac {5}{6}}}$
${\displaystyle \gamma ^{2}-\kappa \leq {\frac {186}{125}}}$

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

${\displaystyle \kappa -\gamma ^{2}-1\geq 0}$

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)