an:04174145
Zbl 0713.62047
Golubev, Grigori K.; Nussbaum, Michael
A risk bound in Sobolev class regression
EN
Ann. Stat. 18, No. 2, 758-778 (1990).
0090-5364 2168-8966
1990
j
62G07 62G20 62C20
integrated mean square error; nonnormal case; lower asymptotic risk bound; nonparametric regression; asymptotic minimax; smoothness ellipsoid; location model; shrinking Hellinger neighborhoods; adaptive bandwidth choice; experimental design; robust smoothing; L2-risk; minimax IMSE; independent error variables; Sobolev-space; bounded fourth moment; optimality; linear estimates; localized bounds; adaptive smoothing
The authors investigate the minimax IMSE for nonparametric estimates of a regression function f in a Sobolev space. The regression model is: \(Y_ i=f(t_ i)+\epsilon_ i\), \(1\leq i\leq n\), with a fixed design \((t_ i)\) distributed according to a density g on [0,1] and with independent error variables \(\{\epsilon_ i\}\) with \(E(\epsilon_ i)=0\). In their main result they give sharp lower bounds for
\[
\lim_{n\to \infty}\inf_{\hat f}\sup_{f,\Pi}n^{2m/2m+1} E_{\Pi}\| \hat f_ n-f\|^ 2_ 2,
\]
where the infimum is taken over all estimates \(\hat f\) and the supremum is taken over all f in the Sobolev-space \(W^ m_ 2(P)\) and all probability distributions \(\Pi\) of \((\epsilon_ 1,...,\epsilon_ n)\) with components which are in a shrinking neighborhood of a fixed distribution and have bounded fourth moment. It is shown that \(\Delta \geq c(m,\sigma^ 2,g,P)\) with an explicit constant c.
This generalizes the case of normal error variables which was treated by \textit{M. Nußbaum} [ibid. 13, 984-997 (1985; Zbl 0596.62052)]. Furthermore, the optimality of c, linear estimates, localized bounds and adaptive smoothing are discussed.
U.Stadtmüller
0596.62052