ENH: stats: make distribution instances picklable

[ev-br]

Intends to close gh-4816.

(this likely has a milestone "after 0.16 is done")

Note to self: multivariate distributions are likely OK, but it's worth checking.

[rgommers]

It passes (except for pep8), but indeed I agree it would be better to merge this after the release split.

[ev-br]

rebased.

[rgommers]

Question: is it a problem that this loses changes to attributes that are made after instantiation (_ctor_param is never updated)? Not sure if there's a good use-case for doing that, but it may be something worth checking for.

[ev-br]

Here's _ctor_param:

        self._ctor_param = dict(
            momtype=momtype, a=a, b=b, xtol=xtol,
            badvalue=badvalue, name=name, longname=longname,
            shapes=shapes, extradoc=extradoc, seed=seed)

a and b are either fixed or get regenerated at each call for some distributions, seed is related to the random_state which is taken care of separately here, the rest do not seem like something a user is very likely to change.

I also note that freezing distributions also relies on _ctor_param.

[rgommers]

I noticed indeed that freezing does the same. It's odd though that there are a bunch of attributes that we aren't preventing users from changing, but assuming that they don't.

It would be easy to add a private method to update self._ctor_param and call that from freeze and from __getstate__. That would solve the issue.

[ev-br]

It would be easy to add a private method to update self._ctor_param and call that from freeze and from __getstate__. That would solve the issue

Done.

[rgommers]

Thanks Evgeni, merged.