Stratified rejection and squeeze method for generating beta random numbers

Generating beta variates via patchwork rejection

“…The fastest existing procedures for generating beta(p, q) variates with p > i and q > 1 are algorithms B4PE (see Schmeiser and Babu [11]) and Bll, part of algo-rithm BSA dealing with the case p > 1, q > 1 (see Sakasegawa [10]). Sakasegawa's method is based purely on acceptance-rejection, therefore it requires at least two uniforms per beta variate.…”

“…Let x~z rain) be the smallest Xz-value which is feasible in respect to the inequality (10). It is plausible that x~2 rain) is a good choice for the abscissa of PzIn fact, it was established numerically that the sum A(x2) = At(x2) + AR(X2) of the tail area AT(X2)= S2x2-"f(x)dx and the (shaded) rejection area AR(x2)= ~,2_,, (2f(x2) -f(2x2 -x) -f(x))dx increases monotonically between xtz rain) and m except for small a below 2.5.…”

“…The resulting algorithm BB is rather simple, has low set-up costs and reasonable speed. Schmeiser and Babu [11] and Sakasegawa [10] utilized piecewise linear envelopes in the center, and exponentials in the tails. Their sampling routines B4PE and B11 are involved, but they are fast at the expense of a more time-consuming set-up.…”

“…Uniformly fast algorithms (i.e. algorithms with bounded computation times for all p > 1, q > 1) have been established in [5], [10], [11]. Cheng [5] applied a log-logistic envelope to the beta distribution of the second kind.…”

“…The algorithms discussed in this article deal with unimodal densities only. For the other cases p < 1 and/or q < 1 we refer to the papers of Atkinson [2], Atkinson and Whittaker [3], [4], Cheng [5], J6hnk [8] and Sakasegawa [10].…”