Geometrizing Rates of Convergence, II

Maximal Uniform Convergence Rates in Parametric Estimation Problems

“…. , y n ) , they use the absolute variation metric (L 1 norm) 11 d n (θ, θ ) = y |f (y; θ) − f (y; θ )|dy, and show that, if a δ −1 n consistent estimator exists, then, for each θ ∈ Θ and every > 0, there exists a positive number t 0 such that, for any t ≥ t 0 ,…”

Maximal uniform convergence rates in parametric estimation problems

“…Example 2 (continued): Prakasa Rao (1968) shows that, for α = 1 and 0 < β < 1/2, the maximum likelihood estimator for the location parameter θ converges at the (inverse) 11 Donoho and Liu (1991b) also treat some nonlinear cases: estimating the rate of decay and the mode of a density, and robust nonparametric regression.…”

Renormalizing Experiments for Nonlinear Functionals

“…Here we only reect on \general" ideas true to the Le Cam spirit. The proof uses a rescaling argument similar to Theorem 1 of [4] The following lower bound is along the lines of [2,3].…”

“…For various types of measurements Y n = ( y 1 ; y 2 ; : : : ; y n ), problems of this form arise in statistical settings, such as nonparametric density estimation and nonparametric regression estimation; but they also arise in signal recovery and image processing. In such problems, there generally exists an \optimal rate of convergence": the minimax risk from n observations, R(n) = infT sup f2F E(T(Y n ) T(f)) 2 , tends to zero as R(n) n r . There is an extensive literature on the determination of such optimal rates for a variety of functionals T, function classes F, and types of observation Y n ; the literature is really too extensive to list here, although we mention [7], [15], and [16].…”

“…Lucien [11] has contributed directly to this literature, in his typical abstract and profound way; his ideas have stimulated the work of others in the eld, e.g. [2].…”