Acceptance-rejection sampling from the conditional distribution of independent discrete random variables, given their sum

The problem of drawing samples from the conditional distribution of a vecto r of independent, but not necessarily identically distributed, discrete ran dom variables, given their sum, is considered. The traditional method of ac ceptance - rejection sampling can be impossible in practice if the statisti c to condition upon has an extremely low probability of assuming the condit ioning value. It is shown how the sufficiency principle combined with tilti ng of distributions may be used to find an equivalent problem, where the co nditioning event has a much higher probability to occur. As applications, a test for proportional hazards in survival analysis and a test of no trend in a legit model with offset are given.