VarBacktest

Coverage tests for a one-day VaR forecast against a realised P&L series: the count of exceptions (a loss worse than the forecast), Kupiec's unconditional proportion-of-failures test, and Christoffersen's independence test plus their combined conditional-coverage statistic.

The reference distributions are chi-square with one and two degrees of freedom, whose survival functions are closed forms — 2 (1 - Phi(sqrt s)) for one degree, exp(-s/2) for two — so no incomplete-gamma routine is needed.