@fkiraly, my apologies, this is basically a copy paste from the normal currently. I have only changed the pdf function at the moment. Still working through the rest. I should have made this clearer.

just posting this here in case you're not aware
https://xkcd.com/1347/

@fkiraly, I couldn't find a formula for energy, if you have one I am happy to implement it. Also, from what I have read it sounds like there isn't an analytic solution for probability point function for the t-distribution. Are you aware of an analytic solution for the ppf?

I couldn't find a formula for energy, if you have one I am happy to implement it.

Thanks! It's nasty though, so don't do it if it feels to unpleasant.

Appendix A.2 of "evaluating forecasts with scoringRules" has a few explicit formulae for the energy, including the t-distribution. The expression is hard to track in implementation, so I would advise comparing against the Monte-Carlo default if you implement it.

Are you aware of an analytic solution for the ppf?

The inverse t cdf tends to appear, here's what Wolfram Alpha says: https://www.wolframalpha.com/input/?i=inverse+cdf+student%27s+t+distribution

here's what Wolfram Alpha says: https://www.wolframalpha.com/input/?i=inverse+cdf+student%27s+t+distribution

Just added the ppf using the Wolfram Alpha link. Thanks for sharing that link.

Have you checked that it's indeed inverse to the cdf?

This should probably be part of the tests, too, note to self.

Have you checked that it's indeed inverse to the cdf?

Here is an example showing the output of ppf is the inverse of cdf.

>>> import pandas as pd
>>> from sktime.proba.t import TDistribution
>>> n = TDistribution(mu=0, sigma=1, df=[[1, 1], [1, 1], [1, 1]])
>>> p = pd.DataFrame({"col1": [0.1, 0.3, 0.5], "col2": [0.7, 0.9, 1.0]})
>>> n.cdf(n.ppf(p))
   col1  col2
0   0.1   0.7
1   0.3   0.9
2   0.5   NaN

@fkiraly, I am unsure where to start with the energy function. I see the section for student's t-distribution in A.2 of "evaluating forecasts with scoringRules" but, since there isn't a flashing neon sign saying "this is the energy function!" I am unsure of what needs to be implemented.

I assume the continuous ranked probability score (CRPS) is related to energy but it is not obvious to me how.

Would it be possible to get a review and perhaps merge in what is currently implemented and leave energy function for a different PR unless you want to take a look at energy on this PR?

Would it be possible to get a review and perhaps merge in what is currently implemented and leave energy function for a different PR unless you want to take a look at energy on this PR?

Sure!

Have you checked why the tests are failing?
There's something going on with indexing. Happy to have a more detailed look if you're not sure what is wrong here.

I assume the continuous ranked probability score (CRPS) is related to energy but it is not obvious to me how.

Yes, the general formula is CRPS(d.cdf, y) = d.energy(y) - 0.5 * d.energy(). Unfortunately the reference only gives CRPS, but the scaling behaviour of the terms (d.energy() does not depend on y and is proportional to d.std()) makes me think that the positive terms are d.energy(y), and then negative term is - 0.5 * d.energy(). If not, d.energy(y) should be the positive terms minus a constant multiple of the negative term (the coefficient should be "simple" and guessable by comparison with the Monte Carlo estimate).

As said, we can leave it for now - no worries. It's unpleasant as I alluded to above (and in addition, unfortunately, not 100% clear from the reference).

Btw @Alex-JG3, thanks a lot!
What would also be useful is feedback on the implementer experience - e.g., is there boilerplate that is unclear, or interface elements that you think would be better changed?