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Chi-squared distribution $\chi^2(k)$ has parameter $k$.

On the one hand, $k$ should be the shape parameter because chi-squared distribution is a special case of Gamma distribution: $\chi^2(k) \equiv \mathrm{Gamma}(k/2, 2)$ and the first parameter of Gamma distribution is called shape. Also changing of $k$ really alters its shape.

On the other hand, wikipedia says that the mean is a location parameter:

Examples of location parameters include the mean, the median, and the mode.

And $k$ is the mean of the chi-squared distribution.

But wiki also says that

Thus in the one-dimensional case if the location parameter is increased, the probability density or mass function shifts rigidly to the right, maintaining its exact shape.

In our case changing of $k$ doesn't shift the PDF but alters its shape as I said above.

So what is the type of parameter $k$: shape parameter or location parameter?

And if $k$ is the shape parameter does it imply that the mean of the distribution is not always its location parameter?

Rodvi
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    Although the formula in the Wikipedia article you quote is correct, the text is plainly wrong. Trust the formula: can you express a $\chi^2(k)$ distribution function in the form $f(x-k)$? (The answer should be obvious: consider the range of values to which any $\chi^2$ distribution assigns nonzero probability density.) Once again, plotting these distributions (as I advised in [a comment to your first question](https://stats.stackexchange.com/questions/340418/types-of-parameters-of-a-parametric-family-of-probability-distributions)) will be revealing. – whuber Apr 14 '18 at 17:27
  • @whuber Well, thanks, I can't express $\chi^2$ pdf in the form $f(x-k)$, therefore $k$ is the shape parameter. But what part of the wikipedia text is wrong: "examples of location parameters include the mean, the median, and the mode." or "if the location parameter is increased, the pdf shifts rigidly to the right, maintaining its exact shape" ? – Rodvi Apr 14 '18 at 17:41
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    The former is incorrect, as your question implicitly points out. The latter is correct but limited to distributions with PDFs. It is also ambiguous in its use of the word "shape," which in this context seems to refer to the familiar notion of shape as defined by similarity transformations (in the plane) rather than as determined by cosets of location-scale equivalences of distributions. Although the distinction I am making is subtle, it is conceptually important and therefore the Wikipedia article's misuse of "shape" here is regrettable. – whuber Apr 14 '18 at 17:50
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    @whuber Great, thank you once again. I asked this question due to the above metioned wrong wikipedia statement about examples. I think wiki confused "location parameter" with "[measure of location (central tendency)](https://en.wikipedia.org/wiki/Summary_statistics)". The measure of location indeed can be the sample mean, the sample median or the sample mode. – Rodvi Apr 14 '18 at 18:13

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