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Abstract
In this work we investigate the relationship among three classical sampling techniques:
the inverse of density (Khintchine's theorem), the transformed rejection (TR) and
the generalized ratio of uniforms (GRoU). Given a monotonic probability density function
(PDF), we show that the transformed area obtained using the generalized ratio of uniforms
method can be found equivalently by applying the transformed rejection sampling approach
to the inverse function of the target density. Then we provide an extension of the
classical inverse of density idea, showing that it is completely equivalent to the
GRoU method for monotonic densities. Although we concentrate on monotonic probability
density functions (PDFs), we also discuss how the results presented here can be extended
to any non-monotonic PDF that can be decomposed into a collection of intervals where
it is monotonically increasing or decreasing. In this general case, we show the connections
with transformations of certain random variables and the generalized inverse PDF with
the GRoU technique. Finally, we also introduce a GRoU technique to handle unbounded
target densities.