Locally minimax efficiency of nonparametric estimates of square-integrable densities

Locally minimax efficiency of nonparametric density estimators for χ2-type losses

“…[21], Lemma 5). Let ~ be a Gaussian random vector with zero mean and covariance matrix S = S(N) E Sn satisfying (2.11) and (3.5).…”

“…The proof follows directly from the argument of Lemma 6 in [21] with the corresponding changes in notation (and therefore is omitted).…”

“…Tsybakov [23] has shown that the result also holds for nonsquared losses, o(f, g) = w(llf -gl [2), and can also be applied when estimating derivatives. Rudzkis and Radavi~ius [21 ] considered the subsets of L2 defined via accuracy of their approximation by the finite-dimensional subspaces (see also [19]). This approach was proposed by Cencov [3], [4].…”

Lower bound for quadratic losses of estimation of infinite-dimensional parameter

“…The case X N ------(X 1 ..... XN) , where X~ ..... XN are independent and identically distributed random vectors with unknown distribution density 0 E 6) C L2(K, Ix), the compact K C R k, and Ix is a finite measure, has been considered in [7,8]. A construction of the estimator 0* which achieves the asymptotic lower bound (1.1) for all 0 E 6)0 cl 6)~ = O, is also presented there.…”

“…Following the scheme of Golubev's paper [4] we extend the lower bound of locally minimax risk with quadratic losses obtained in [7,8] for i.i,d, observations to the general class of statistical experiments.…”