ENH: special: add log of wright_bessel

[lorentzenchr]

Is your feature request related to a problem? Please describe.

When the log-likelihood of a Tweedie distribution (compound Poisson-Gamma) is computed, one needs to evaluation np.log(wright_bessel(a, 0, x)) where 0<a<inf and x is potentially large. This can lead to inf or nan value, see statsmodels/statsmodels#9234.

Describe the solution you'd like.

Provide a new function log_wright_bessel that takes care for large x. Similar functions are

• log_ndtr
• log_expit
• loggamma

Describe alternatives you've considered.

No response

Additional context (e.g. screenshots, GIFs)

The implementation would be relatively straight forward:

• wb_small_a: Replace multiplicative exp(x) * rgamma(b) by +x-loggamma(b)
• wb_large_a: similar
• wb_asymptotic: replace multiplicative std::pow(Z, 0.5 - b) * std::exp(Ap1[1] / a * Z) by +(0.5 - b) * log(Z) + Ap1[1] / a * Z

This would already work for most cases.

wright_bessel was introduced in v1.7.0 with PR #11313.