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This paper continues the research started in \cite{LW16}. In the framework ofthe convolution structure density model on $\bR^d$, we address the problem ofadaptive minimax estimation with $\bL_p$--loss over the scale of anisotropicNikol'skii classes. We fully characterize the behavior of the minimax risk fordifferent relationships between regularity parameters and norm indexes in thedefinitions of the functional class and of the risk. In particular, we showthat the boundedness of the function to be estimated leads to an essentialimprovement of the asymptotic of the minimax risk. We prove that the selectionrule proposed in Part I leads to the construction of an optimally or nearlyoptimally (up to logarithmic factor) adaptive estimator.

Estimation in the convolution structure density model. Part II: adaptation over the scale of anisotropic classes