[DOC] docstring with mathematical description for QPD_Empirical

[fkiraly]

This PR adds a mathematical description in the docstring of QPD_Empirical.

The docstring also explains why the distribution is quantile parameterized, and its relation to Empirical.

A review would be appreciated, @Ram0nB, @setoguchi-naoki, @FelixWick - I also hope (and would appreciate a check) for typos in the math.

FYI @VascoSch92, as this is an example where two formal objects:

• are equivalent if considered without parameterization
• but not equivalent if considered together with the parameterization

and the more general question what the class/design principle should be here. Compare Fibonacci vs OEIS(n) object. Also relates to the "multiple names" discussion in VascoSch92/sequentium#49, I am currently of the opinion that different class names should imply difference in parametric object.

[VascoSch92]

Just one question:
you say: such that w_i = (p_{i+1} - p_{i-1})/2 for 1 = 1, \dots, N and then you define p_0 and p_{N+1}. Why you define them? How are these two values playing a role here?

[fkiraly]

$w_i$ is defined in terms of $p_{i+1}$ and $p_{i-1}$ for $i= 1\dots N$, and $p_i$ have been defined above also only for $i= 1\dots N$.

So we need to treat $w_1$, $w_N$, either by writing them down explicitly, or by writing equivalent values for $p_0$, $p_{N+1}$. I thought the latter was clearer, in terms how the boundary treatment looks like?

[VascoSch92]

Ah sorry now I see ;-)

thanks for the explanation. I didn't check that there it was a p_{i-1}) and we have to define p_0 (same for p_{i+1}

[fkiraly]

the $w_i$ should sum to 1, hopefully. The weights should make it that the steps in the ppf are exactly in the middle between two adjacent $p_i$.