Generating Random Samples from Non-Identical Truncated Order Statistics

We provide an efficient algorithm to generate random samples from the bounded kth order statistic in a sample of independent, but not necessarily identically distributed, random variables. The bounds can be upper or lower bounds and need only hold on the kth order statistic. Furthermore, we require access to the inverse CDF for each statistic in the ordered sample. The algorithm is slightly slower than rejection sampling when the density of the bounded statistic is large, however, it is significantly faster when the bounded density becomes sparse. We provide a practical example and a simulation that shows the superiority of this method for sparse regions arising from tight boundary conditions and/or over regions of low probability density.