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There exist several tests of normality. This is, tests for checking if a data set is normally distributed.

Is there a "Test of Weibullity"? This is, a test for checking if a data set is Weibull distributed.

kjetil b halvorsen
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    One-sample Kolmogorov-Smirnov can be used as a test of a Weibull distribution. [The usual concerns about how useful this kind of distribution testing is apply, however.](https://stats.stackexchange.com/questions/2492/is-normality-testing-essentially-useless) – Dave Jul 26 '21 at 14:42
  • @Dave The limitation of that test is that it only works with fixed parameter values. So, it is about whether the distribution is Weibull with specific parameters, not about whether the distribution belongs to the Weibull family. – Golovkin Jul 26 '21 at 14:48
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    A simple Q-Q-plot could be used as a visual diagnostic display. But you can never "prove" that the data comes from the Weibull family because there are infinitely many other distributions that are compatible with the data at the same time. All you can hope for is to show a reasonable degree of compatibility with a Weibull population. – COOLSerdash Jul 26 '21 at 14:52
  • @COOLSerdash The QQ plot suggestion is very useful, thanks. I am not sure about your last comment as this is indeed what normality tests try to do, don't they? – Golovkin Jul 26 '21 at 15:12
  • No. @Dave linked to a really helpful thread about the matter which I encourage you to read (see for example AdamO's answer). – COOLSerdash Jul 26 '21 at 15:16
  • @COOLSerdash I am surprised you deny that is the aim of normality tests. I understand how easily they fail in real life, but they are well defined theoretical questions. Most of the questions in that thread are about how difficult real life is, but not about a formal proof of the incorrectness of normality tests. – Golovkin Jul 26 '21 at 15:20
  • Normality tests are not incorrect, but they can't prove that the data is normally distributed. They test the null hypothesis that the data comes from normal population. Insofar, they can only provide evidence against this null hypothesis. A failure to reject this null hypothesis does *not* mean that the data come from a normal distribution. It shows that they are compatible with a normal distribution. As mentioned: They are compatible with infinitely many other distributions at the same time. – COOLSerdash Jul 26 '21 at 15:25
  • @COOLSerdash Yes, but that is the case of any hypothesis test, not just normality tests, which is the main point of my question. – Golovkin Jul 26 '21 at 15:35
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    You can use the Lilliefors variant of K-S. Regardless, the cautions about distribution testing are well advised. What is the matter with just examining a suitable probability plot so you can learn precisely *how* your data might depart from appearing Weibull? That usually is more insightful and productive. – whuber Jul 26 '21 at 17:43
  • See https://www.researchgate.net/publication/325094819_Weibullness_test_and_parameter_estimation_of_the_three-parameter_Weibull_model_using_the_sample_correlation_coefficient – kjetil b halvorsen Jul 27 '21 at 12:06

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