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Following my earlier post on Kindermanâ€™s and Monahanâ€™s (1977) ratio-of-uniform method, I must confess I remain quite puzzled by the approach. Or rather by its consequences. When looking at the set A of (u,v)â€™s in R?×X such that 0?u²?ƒ(v/u), as discussed in the previous post, it can be represented by its parameterised boundary

u(x)=?ƒ(x),v(x)=x?ƒ(x) x in X

Similarly, since the simulation from ƒ(v/u) can also be derived [check Luc Devroyeâ€™s Non-uniform random variate generation in the exercise section 7.3] from a uniform on the set B of (u,v)â€™s in R?×X such that 0?u?ƒ(v+u), on the set C of (u,v)â€™s in R?×X such that 0?u³?ƒ(v/?u)², or on the set D of (u,v)â€™s in R?×X such that 0?u²?ƒ(v/u), which is actually exactly the same as A [and presumably many other versions!, for which I would like to guess the generic rule of construction], there are many sets on which one can consider running simulations. And one to pick for optimality?! Here are the three sets for a mixture of two normal densities:

For instance, assuming slice sampling is feasible on every one of those three sets, which one is the most efficient? While I have no clear answer to this question, I found on Sunday night that a generic family of transforms is indexed by a differentiable monotone function h over the positive half-line, with the uniform distribution being taken over the set