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?
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$\begingroup$ 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 $\endgroup$– argyrisCommented Dec 8, 2013 at 17:32
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2$\begingroup$ I wouldn't say the uniform was unimodal. It doesn't satisfy your definition. $\endgroup$– Nick CoxCommented Dec 8, 2013 at 17:40
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$\begingroup$ @argyris: You migh have got me wrong: I recalled these distributions as an example for unimodal ones. $\endgroup$– ChrisCommented Dec 9, 2013 at 13:03
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$\begingroup$ @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. $\endgroup$– ChrisCommented Dec 9, 2013 at 13:07
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2$\begingroup$ 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) $\endgroup$– Glen_bCommented Dec 9, 2013 at 22:01
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1 Answer
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You're asking two questions:
- Is there a generic test for unimodality?
- 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.
