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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.
Reviewer: U.Stadtmüller

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62C20 Minimax procedures in statistical decision theory
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