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Let $X$ be a continuous random variable which induces a probability measure on $\mathbb{R}$ denoted by $\mu$. Are there any instances in statistics when we deal with random variables $X$ such that $\frac{d\mu}{d\lambda}$ does not exist (where $\lambda$ is the usual Lebesgue measure)?

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    $\begingroup$ A continuous random variable is a random variable whose cumulative distribution function is continuous everywhere, this does not imply it is differentiable anywhere. See, e.g., the Cantor function. $\endgroup$
    – Xi'an
    Commented Dec 10, 2022 at 10:26
  • $\begingroup$ See also stats.stackexchange.com/a/585601/7224 $\endgroup$
    – Xi'an
    Commented Dec 10, 2022 at 10:34
  • $\begingroup$ @Xi'an That was not my question, " are there instances in statistics " , not in mathematics in general, i.e. are such distributions ever used in statistical analysis? $\endgroup$ Commented Dec 10, 2022 at 15:01
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    $\begingroup$ I do not have an example in dimension one, but some copulas are not everywhere differentiable, see the above link. $\endgroup$
    – Xi'an
    Commented Dec 10, 2022 at 17:27
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    $\begingroup$ @Xi'an That is a good example thank you. $\endgroup$ Commented Dec 10, 2022 at 17:45

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An obvious example is a singular multinormal distribution, which does not have a density with respect to Lebesgue in the full space. This occurs very frequently in multivariate statistics, and for example as the distribution of residuals in multiple linear regression.

Another example, from circular statistics, is the uniform distribution on the circle or sphere, which does not have densities with respect to Lebesgue in the full ambient space. So there is no lack of examples!

A simple example is the uniform distribution on the line $y=x$ for $0\le x, y \le 1$, which has cdf $F(x,y)=\min(x,y)$. For other examples see What does it mean for a probability distribution to not have a density function? and references there.

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