The patchwork rejection technique for sampling from unimodal distributions

Reviewer: R. Sambasiva Rao

The rejection regions for many continuous and discrete statistical distributions are available in the literature. This paper is an outcome of the authors' continuous research work [1–4] in sampling methods. Minh [5] proposed a special sampling method for gamma distribution (using the hat function and minorizing rectangles) that rearranges the area under the histogram so that the samples may be generated within a large central interval. Exponential functions cover the tails of the function. Kemp [6] introduced improvement in the acceptance/rejection procedures applicable to continuous distributions with fixed parameters. Recently, universal rejection algorithms from log-concave discrete distributions and automatic generators have been reported. It is interesting that all of these methods have uniformly bounded sampling time. Stadlober has previously reported [1,2] on the algorithm and software for generating discrete random variates and sampling from Poisson, binomial, and hypergeometric distributions. Here, he reports on the patchwork rejection technique. The paper is divided into four sections and one appendix. The introduction deals with the status of sampling methods of acceptance-rejection type for several statistical distributions. In section 2, a general procedure for patchwork technique is described for continuous and discrete distributions, and beta binomial/Poisson distributions are considered. The authors have incorporated a pseudocode method for sampling discrete distribution in section 3. Section 4 is devoted to comparisons of algorithms and software implementation in Fortran and Turbo C++. The appendix contains proofs for feasible reflection centers and applications to hypergeometric distribution. The patchwork rejection is confined here to unimodal distributions. The area below the histogram is rearranged in the center. The PRD algorithm is applied to Poisson, binomial, and hypergeometric distributions. A staircase function is added for fast sampling, and the code is implemented in C in the WinRand 1.0/95 package, accessible by anonymous ftp from statistic.tu-graz.ac.at/winrand. The method described here is compared with other available methods in terms of speed and memory. For hypergeometric distributions, the current method is 1.3 to 2.5 times faster than the uniform exponential rejection (H2PE) and 1.2 to 3 times faster than the ratio of uniforms (HRUE).