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The paper deals with the problem of nonparametric estimating the $L_p$--norm,$p\in (1,\infty)$, of a probability density on $R^d$, $d\geq 1$ fromindependent observations. The unknown density %to be estimated is assumed tobelong to a ball in the anisotropic Nikolskii's space. We adopt the minimaxapproach, and derive lower bounds on the minimax risk. In particular, wedemonstrate that accuracy of estimation procedures essentially depends onwhether $p$ is integer or not. Moreover, we develop a general technique forderivation of lower bounds on the minimax risk in the problems of estimatingnonlinear functionals. The proposed technique is applicable for a broad classof nonlinear functionals, and it is used for derivation of the lower bounds inthe~$L_p$--norm estimation.

Minimax estimation of norms of a probability density: I. Lower bounds