Minimax Bound on the Risk of Nonparametric Density Estimates

The author obtains a minimax lower bound, exact in order of magnitude as $n\to\infty$, for moments $\parallel\gamma_n-f\parallel$, where $f(\cdot)$ is an unknown density from some class of sufficiently smooth functions, , $\gamma_n(\cdot)$ is the density estimate, and $\parallel\cdot\parallel$ is the norm in $L_2(-\infty,\infty)$. The results are allied to those of Chentsov and Farrell [N. N. Chentsov, Statistical Decision Rules and Optimal Conclusions (in Russian), Nauka, Moscow, 1972; R. Farrell,cAnn. Math. Statist., 1972, vol. 43, no. 1, pp. 170–180].