  Traditionally, the Method of (Shannon-Kullback's) Relative Entropy
Maximization (REM) is considered with linear moment constraints. In this work,
the method is studied under frequency moment constraints which are non-linear
in probabilities. The constraints challenge some justifications of REM since a)
axiomatic systems are developed for classical linear moment constraints, b) the
feasible set of distributions which is defined by frequency moment constraints
admits several entropy maximizing distributions (I-projections), hence
probabilistic justification of REM via Conditioned Weak Law of Large Numbers
cannot be invoked. However, REM is not left completely unjustified in this
setting, since Entropy Concentration Theorem and Maximum Probability Theorem
can be applied.
  Maximum Renyi-Tsallis' entropy method (maxTent) enters this work because of
non-linearity of X-frequency moment constraints which are used in Non-extensive
Thermodynamics. It is shown here that under X-frequency moment constraints
maxTent distribution can be unique and different than the I-projection. This
implies that maxTent does not choose the most probable distribution and that
the maxTent distribution is asymptotically conditionally improbable. What are
adherents of maxTent accomplishing when they maximize Renyi's or Tsallis'
entropy?
