ENH transforms: symmetric inverse of u-shaped transforms

[josef-pkt]

In sandbox I included hump- and u-shaped transforms like square mapping form R to R+. However, I don't have inverse functions for those transforms.

in Links, we have power transform restricted to R++, which also work for interval [0, 1]

One possibility to invert square is to construct a symmetric distribution on R, by allocating sqrt equally to both signs +/- .

example Kotz distribution is sqrt of chi2(df=2 * rho - 1) with equal prob for + and - sign
equ. (2) and (3) in

Salehi, Mahdi, and Adelchi Azzalini. 2018. “On Application of the Univariate Kotz Distribution and Some of Its Extensions.” METRON 76 (2): 177–201. https://doi.org/10.1007/s40300-018-0137-3.

I guess we can just do compounding/mixing of an R+ distribution with "sign" distribution, prob(-1) = prob(1) = 0.5

possible use: compute cdf/ppf through a "known" or "special" distribution
e.g. mixing with sign distribution, needs to stitch together the parts on positive and negative half line. (I guess using sf and cdf)
scipy uses special functions for chi and chi2, cdf, ppf.

[josef-pkt]

Kotz family looks like a useful elliptical distribution if we can get univariate cdf and ppf.