`stats.norminvgauss` incorrect implementation?

Hi,

Mentioned in the latest release of TensorFlow Probability, it has been noted that scipy.stats.norminvgauss has not been implemented in the same way as R or the original mathematical paper.

It seems the loc and scale parameters should actually be mu and delta. Is there a way to update this so that non-standard cases (other than loc=0, scale =1) make mathematical sense?

Note that `scipy.stats.norminvgauss` implements the distribution as a
      location-scale family. However, in the original paper, and other
      implementations (such as R) do not implement it this way. Thus the
      `scale` parameters here and scipy don't match unless `scale = 1`.
  Warning: As mentioned above, this distribution is __not__ a location-scale
  family.

Link here: https://github.com/tensorflow/probability/blob/main/tensorflow_probability/python/distributions/normal_inverse_gaussian.py

Docs: https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/NormalInverseGaussian

[Nishan-Pradhan]

@mdhaber Thanks! That works really well!

[mdhaber]

Very close, but don’t use norminvg to calculate the PPF and ISF. The functions will only be evaluated at x = tol/10, so you can code in correct values yourself. It is not very sensitive to these values being super accurate - e.g., for PPF, return any value x at which norminvg.cdf(x) is very small (ideally tol/10).

Something like

class CustomNIG:

    def pdf(self, x):
        return norminvg.pdf(x)

    def cdf(self, x):
        return norminvg.cdf(x)

    def ppf(self, x):
        return -2  # norminvg.cdf(-2) ~ 0

    def isf(self, x):
        return 2  # norminvg.sf(2) ~ 0


dist = CustomNIG()
norminvg_fast = stats.NumericalInverseHermite(dist, tol=1e-6)

ppf = norminvg_fast.ppf(data)

This will make NumericalInverseHermite form an approximation H of the inverse of norminvg.cdf that is valid between -2 and 2. The round-trip tolerance (i.e. abs(x - norminvg.cdf(H(x)))) will be controlled at mesh points and midpoints between mesh points to tol=1e-6.