CDF and PPF accuracy

Gamma and inverse-gamma quantiles use the analytical distribution function. Inverse-Gaussian and interior GIG quantiles use a trapezoidal PINV table (HormannLeydold2011). A degenerate GIG delegates to the Gamma or inverse-gamma CDF.

Tolerances

These are the agreements the test suite pins against SciPy.

Family	Method	Tolerance
Gamma, inverse-gamma	analytical	rtol \(10^{-6}\), atol \(10^{-8}\)
Inverse-Gaussian, interior GIG	4000-point table	rtol \(10^{-3}\), atol \(10^{-4}\)
GIG round trip cdf(ppf(q))	same table	rtol \(10^{-4}\), atol \(10^{-5}\)
Degenerate GIG	Gamma or inverse-gamma limit	rtol \(10^{-8}\)

The grid has 4000 points. Tail bisection stops at mass \(10^{-14}\). Left of the grid, cdf returns 0. Right of the grid, cdf returns 1. ppf interpolates on the stored quantile nodes.

The interpolant is differentiable in \(x\). A derivative in the parameters differentiates the grid too. That derivative belongs to this approximation, not to a separate exact CDF. Call quantile_table() once when many quantiles share the parameters. cdf and ppf are JIT-safe. The degeneracy test uses lax.cond.

The CPU and JAX paths both evaluate log_kv inside normix. SciPy kve is a test oracle, not the CDF.