[ENH] distributions: add missing explicit energy computations

[fkiraly]

A number of existing distibutions have no energy formula implemented. In many cases, it is possible to obtain an analytic or numerically explicit formula, while the default delegates to a much less efficient Monte Carlo approximation.

Recipe:

1. find any disttibution without energy or _energy_ etc implementations
2. look up or derive the energy formula
3. implement in the distirbution
4. check against the Monte Carlo default, these should be close.

How to obtain the energy formula, there are multiple options:

• Appendix A.2 of "evaluating forecasts with scoringRules" has a few explicit formulae for the CRPS (see below on how to obtain energy from CRPS)
• ngboost has CRPS classes implemented for most distributions it implements: https://github.com/stanfordmlgroup/ngboost/tree/master/ngboost/distns (see below on how to obtain energy from CRPS)
• using Wolfram Alpha or equivalent symbolic integration tools, on one of the integrals (from expectations), using explicit form of pdf or cdf
• using relationship with truncated distributions and their expectations, see here: [ENH] roadmap of probability distributions to implement #22 (comment)

Some partial derivations and discussions can also be found here:
#219
#220

Also, some of the above sources do not give energy, but CRPS - the two are related, by definition as

$$d.\mbox{CRPS}(y) = \mathbb{E}|X - y| - \frac{1}{2}\mathbb{E}|X - X'|,$$ where $X, X'$ are i.i.d. distributed according to $d$.

From this definition, and basic properties of random variables, one can derive

$$\lim_{y \rightarrow \infty} d.\mbox{CRPS}(y) - y = -\mathbb{E}[X] - \frac{1}{2}\mathbb{E}|X - X'| = -d.\mbox{mean()} - \frac{1}{2} d.\mbox{energy()},$$

i.e., one can obtain the self-term via taking a limit, if the expressions for CRPS and the expectation are already known.

The cross-term can then be obtained by adding the self-term to the CRPS formula,

$$d.\mbox{CRPS}(y) = d.\mbox{energy}(y) - \frac{1}{2}d.\mbox{energy()},$$

using that CRPS and self-term are known, allowing to obtain the cross-term.