New Lower Bounds for Testing Monotonicity and Log Concavity of
  Distributions
Yuqian Cheng Daniel M. Kane Zhicheng Zheng
We develop a new technique for proving distribution testing lower bounds for
properties defined by inequalities involving the bin probabilities of the
distribution in question. Using this technique we obtain new lower bounds for
monotonicity testing over discrete cubes and tight lower bounds for
log-concavity testing.
  Our basic technique involves constructing a pair of moment-matching families
of distributions by tweaking the probabilities of pairs of bins so that one
family maintains the defining inequalities while the other violates them.