Transformed Density Rejection (TDR)

The main problem when utilizing the acceptance-rejection method for designing a random variate generator is the choice of the hat function. In Sect. 2.2 we have demonstrated how we can find a hat function “by hand” for some simple examples. But for an automatic algorithm we need a design method that can be executed by the computer. It is possible to find a single hat that can be used for all distributions of fairly large families but it is obvious that the fit of such a “global hat” cannot be very close for all distributions. In Chap. 6 we are going to discuss how to find and use such global hat functions. In this chapter we are dealing with methods that can construct hat functions automatically around a given point locally, at least if the density satisfies some regularity conditions. Transformed density rejection is such a principle that can be applied to a large class of continuous distributions. A first variant has been published by Gilks and Wild (1992) under the name adaptive rejection sampling. The use of a general transformation and the name transformed density rejection was suggested by Hörmann (1995). In this presentation we try to develop the idea such that the reader can also see the (in our opinion) “beautiful” mathematical background (Sects. 4.2–4.4). The details and variants of the algorithms are presented in Sect. 4.4. Generalizations and special cases are discussed in Sects. 4.6 and 4.7.