  We study the adaptive minimax estimation of non-linear integral functionals
of a density and extend the results obtained for linear and quadratic
functionals to general functionals. The typical rate optimal non-adaptive
minimax estimators of "smooth" non-linear functionals are higher order
U-statistics. Since Lepski's method requires tight control of tails of such
estimators, we bypass such calculations by a modification of Lepski's method
which is applicable in such situations. As a necessary ingredient, we also
provide a method to control higher order moments of minimax estimator of cubic
integral functionals. Following a standard constrained risk inequality method,
we also show the optimality of our adaptation rates.
