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A risk bound in Sobolev class regression. (English) Zbl 0713.62047
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 M. Nußbaum [ibid. 13, 984-997 (1985; Zbl 0596.62052)]. Furthermore, the optimality of c, linear estimates, localized bounds and adaptive smoothing are discussed.