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In this work we investigate the relationship among three classical samplingtechniques: the inverse of density (Khintchine's theorem), the transformedrejection (TR) and the generalized ratio of uniforms (GRoU). Given a monotonicprobability density function (PDF), we show that the transformed area obtainedusing the generalized ratio of uniforms method can be found equivalently byapplying the transformed rejection sampling approach to the inverse function ofthe target density. Then we provide an extension of the classical inverse ofdensity idea, showing that it is completely equivalent to the GRoU method formonotonic densities. Although we concentrate on monotonic probability densityfunctions (PDFs), we also discuss how the results presented here can beextended to any non-monotonic PDF that can be decomposed into a collection ofintervals where it is monotonically increasing or decreasing. In this generalcase, we show the connections with transformations of certain random variablesand the generalized inverse PDF with the GRoU technique. Finally, we alsointroduce a GRoU technique to handle unbounded target densities.

On the Generalized Ratio of Uniforms as a Combination of Transformed Rejection and Extended Inverse of Density Sampling