[ENH] Tweedie distribution, incl mathematics and design

[fkiraly]

For https://github.com/sktime/skpro/issues/423 and other GLM interfaces, we would need a Tweedie distribution.

The Tweedie distribution has a number of difficulties:

• it is parameterized by a power law parameter p, a real number, which leads to special cases that include well-known and obscure distribution families. This also leads to design questions on how these families and Tweedie should relate
• it is mixed, so cannot be represented in the scipy interface, hence it does not exist in scipy for all p. In fact, scipy has no Tweedie distribution, see discussion here: https://github.com/scipy/scipy/issues/11291#issuecomment-1868256070
• the problem of how to best compute various functions for the various obscure Tweedie ED subfamilies is bleeding edge research, even for cdf or ppf

There is some discussion about Tweedie in the issue related to the sklearn Tweedie regressor: https://github.com/sktime/skpro/issues/423#issuecomment-2231637334

[fkiraly]

Regarding point 1, architecture, I would propose a delegator design, using the _DelegatedDistribution:

• we first implement the different subfamilies
• then Tweedie can be a delegator, delegating to the subfamily determined by p
• also, the delegator still has to be changed to delegate private methods, not public - given the new distributions interface. This is a separate matter but a condition for a clean implementation

[fkiraly]

Regarding point 3, section 3 of Withers, Nadarajah - "On the compound Poisson-gamma distribution" has numerically exact approximation formulae for cdf and ppf, for the CPG.

[fkiraly]

@ShreeshaM07, could you kindly reply to this so I can assign you?

[ShreeshaM07]

Sure. So I have been working on the implementation of the Tweedie Distribution. So far completed the pdf and cdf implementation of the sub distribution which is the compound poisson gamma distribution in #428.