  Experimental designs intended to match arbitrary target distributions are
typically constructed via a variable transformation of a uniform experimental
design. The inverse distribution function is one such transformation. The
discrepancy is a measure of how well the empirical distribution of any design
matches its target distribution. This chapter addresses the question of whether
a variable transformation of a low discrepancy uniform design yields a low
discrepancy design for the desired target distribution. The answer depends on
the two kernel functions used to define the respective discrepancies. If these
kernels satisfy certain conditions, then the answer is yes. However, these
conditions may be undesirable for practical reasons. In such a case, the
transformation of a low discrepancy uniform design may yield a design with a
large discrepancy. We illustrate how this may occur. We also suggest some
remedies. One remedy is to ensure that the original uniform design has optimal
one-dimensional projection, but this remedy works best if the design is dense,
or in other words, the ratio of sample size divided by the dimension of the
random variable is relatively large. Another remedy is to use the transformed
design as the input to a coordinate-exchange algorithm that optimizes the
desired discrepancy, and this works for both dense or sparse designs. The
effectiveness of these two remedies is illustrated via simulation.
