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Are there any generic tests to validate if a given sample follows a unimodal distribution, like a Gaussian, Cauchy, Student's t or a chi-square?

kjetil b halvorsen
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Chris
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  • Note that you may think of the Unimodal distribution as a "family" of distributions. The common characteristic is that there is one *mode*. A mode has the property that its probability density is equal to the maximum density of the pdf, while at every direction away from the mode the density is *non-increasing*. Under this definition, all the distributions you mention fall under the unimodal category: *Gaussian, Cauchy, Student's t* or a *Chi-square*. In addition, the *Uniform* distribution is also unimodal. In fact, Uniform is the distribution that Hartigan and Hartigan have used in the [pape – argyris Dec 08 '13 at 17:32
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    I wouldn't say the uniform was unimodal. It doesn't satisfy your definition. – Nick Cox Dec 08 '13 at 17:40
  • @argyris: You migh have got me wrong: I recalled these distributions as an example for unimodal ones. – Chris Dec 09 '13 at 13:03
  • @NickCox: You're right, but Hartigan's test returns a p>0.2 for uniform distributions of sample size > 500. Thus, there is a high risk of judging them to be non-multimodal, although they are in fact considered unimodal. – Chris Dec 09 '13 at 13:07
  • I don't understand your "but" here. Does "non-multimodal" mean anything but "unimodal"? The Hartigan-Hartigan test [let's not forget that it's two of them, husband and wife] treats the uniform as reference as in a sense the worst case unimodal. @argyris' definition doesn't seem to match that. – Nick Cox Dec 09 '13 at 13:15
  • @NickCox: I can hardly accept a uniform as actually "unimodal", if it's more a case of many modes of the same density. Nevertheless, I might be able to accept it as "the worst case unimodal", as you described it. In this case, Hartigans p-Value also makes some sense. – Chris Dec 09 '13 at 13:39
  • I am not sure what hinges on your acceptance here. Your first sentence echoes my first point. My "worst case unimodal" was a paraphrase of the Hartigan-Hartigan approach; their Abstract alone gives a more formal wording. – Nick Cox Dec 09 '13 at 13:46
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    There are numerous tests for unimodality (I think I once identified about nine). I think the dip test is the most well-known. (Edit: [yep - it is nine if you include the Donoho test mentioned at the end](http://stats.stackexchange.com/questions/51062/test-for-bimodal-distribution/51085#51085)) – Glen_b Dec 09 '13 at 22:01
  • NB for anyone considering this a duplicate on the basis of the above link - I would lean toward not marking this question as a duplicate, because even though that answer applies here, the question there is different - the bimodal alternative in the question there is not the same as the 'not unimodal' alternative in the question here. – Glen_b Dec 09 '13 at 22:08
  • @Glen_b: and particular, Hartigans' test is the one implemented in R! :-) – Chris Dec 10 '13 at 12:37
  • @Chris I mention which package it's in at the linked answer above. – Glen_b Dec 10 '13 at 20:08

1 Answers1

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You're asking two questions:

  1. Is there a generic test for unimodality?
  2. Are there tests to test whether a sample is derived from a given distribution, say, a normal distribution?

Ad 1): Yes, the Hartigan-Hartigan dip test, Ann. Statist. 13(1):70-84.

Ad 2): There exists a number of special tests, but the Kolmogorov-Smirnov test is a general-purpose nonparametric test, although with low statistical power.

Best,

abaumann
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  • You are right, sorry for the ambiguity. Thank you very much for your quick and precise answer! – Chris Oct 11 '13 at 09:14