New sampling techniques for calculation of Dirichlet probabilities

In this paper there will be described and compared several simulation algorithms for computing the cumulative distribution function values of Dirichlet distribution. New sampling techniques as Sequential Conditioned Sampling (SCS) and Sequential Conditioned Importance Sampling (SCIS) will be introduced. On the base of an interesting property of the Dirichlet distribution new versions of the SCS and SCIS algorithms will be developed, called SCSA, SCSB and SCISA, SCISB, respectively. SCIS and the modified algorithms need more CPU time but they result significant variance reduction. Their resultant efficiency will be compared to the simple ‘hit-or-miss Monte Carlo’ simulation method with conventional sampling of the Dirichlet distributed random vectors what we call Crude Monte Carlo (CMC) simulation method in the paper. Numerical test results will also be presented.