  We develop a non-negative polynomial minimum-norm likelihood ratio (PLR) of
two distributions of which only moments are known. The sample PLR converges to
the unknown population PLR under mild conditions. The methodology allows for
additional shape restrictions, as we illustrate with two empirical
applications. The first develops a PLR for the unknown transition density of a
jump-diffusion process, while the second extracts a positive density directly
from option prices. In both cases, we show the importance of implementing the
non-negativity restriction.
