MAINT: cauchy moments are undefined

[argriffing]

See #2055 (comment).

This PR could weaken some tests by not failing moment calculations that mistakenly return nan.

[argriffing]

I'm not sure what this comment is about:
# this did not work, skipped silently by nose

[ev-br]

I'm not sure I understand the reasoning in the Wikipedia article linked in #2055 (comment). Using the principal value for the mean looks sensible, and using anything but Cauchy principal value looks strange (to me, FWIW). Anyway, I do not think it's worth worrying about too much, so I'm merging this as is. Thanks Alex and sorry this PR received no attention for this long.

[argriffing]

Using the principle value for the mean looks sensible, and using anything but Cauchy principal value looks strange (to me, FWIW).

Just to make sure I understand, you are suggesting that for any symmetric distribution using the point of symmetry for the mean looks sensible and anything else looks strange?

[josef-pkt]

It might be sensible, but that's the median and not the mean (expected value).

IIRC I have seen also "centrality parameter" for estimates of the center of location for heavy tailed distribution estimation, which is more vague and can be estimated by truncating the sample by ignoring or clipping large absolute values.

[ev-br]

Essentially, yes.
Infinite limits of integration imply some limiting procedure, and if an integrand is symmetric, using $\lim_{A\to\infty}\int_{-A}^{A} (...) dx seems natural, and I fail to see a reason for anything else other than rigor for the sake of rigor. (And it's clear from above that I don't really have a proper math culture :-), so FWIW).

[argriffing]

I fail to see a reason for anything else other than rigor for the sake of rigor.

Here's an imperfect analogy with the definition of prime numbers. It could be possible to redo definitions and theorems in number theory so that 1 is a prime number, but it would require adding nitpicky caveats to the existing theorems like 'every positive integer has unique prime factorization'. Similarly I think that definitions and theorems in probability theory could be redone so that the mean is the principal value mean, but this would requiring adding nitpicky caveats to theorems like the laws of large numbers. I'm not sure if this is correct, but if it is then it could be a better reason than rigor for the sake of rigor.

[josef-pkt]

It's not math or engineering, it's statistics and probability theory

Wikipedia mentions that the law of large numbers doesn't hold in this case

>>> rvs = np.random.standard_t(2, 1000000)
>>> rvs.mean()
0.0015866917880266631
>>> rvs = np.random.standard_t(5, 1000000)
>>> rvs.mean()
0.00059190547465306641
>>> rvs = np.random.standard_t(50, 1000000)
>>> rvs.mean()
0.00078361448965023927

>>> rvs = np.random.standard_t(1, 1000000)
>>> rvs.mean()
-2.4484819996823868

use case:

Theorem

Assume the first 4 moments exist and are finite, then the following holds

blabla

[josef-pkt]

And as practical consequence

The unit tests exclude some checks, or excluded them, when a moment is inf or nan.