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Nonparametric estimation of a regression function. (English) Zbl 0744.62054
Let \((X_ 1,Y_ 1),\dots,(X_ n,Y_ n)\) be i.i.d. on \(I\times\mathbb{R}\), where \(I\) is a compact set in \(\mathbb{R}^ p\). Let \(m(x)=E(Y\mid X=x)\). Let \(F\) be the marginal distribution function (d.f.) of \(X\) and let \(F_ n\) be its empirical d.f. It is assumed that \(\sup_ x E(Y^{4s}\mid X=x)<\infty\) for some integer \(s\geq 2\). Let \(\{W_{nk}:\;k=(k_ 1,\dots,k_ p)\in D_ n\}\) be a sequence of weight functions (depending on \(F\)) on \(I\times I\), where \(D_ n\) is an index set and \(\hbox{card}(D_ n)=K_ n\) with \(K_ n/n^ s\to 0\).
From the above sequence of weight functions the authors construct a sequence of estimates \[ \hat m_ k(x)=\sum Y_ jW_ k(x,X_ j,F_ n)/n,\qquad k\in D_ n. \] A data-dependent method of choosing the (smoothness) index \(k\) which minimizes the prediction square error is proposed. Since this leads to a \(\tilde k\) which depends on an unknown distribution of \((X,Y)\), the authors heuristically motivate applying \(\hat k\) which is the minimizer of \[ \hat T_ n(k)=n^{- 2}\sum\hat\varepsilon^ 2_{kj}[1+2n^{-1}W_ k(X_ j,X_ j,F_ n)], \] where \(\hat\varepsilon_{kj}=Y_ j-\hat m_ k(X_ j)\). Then they use \(\hat m_ k\) as an estimate of the unknown regression function \(m(x)\). This estimate can be specialized to piecewise polynomial, spline, orthogonal series, kernel and nearest neighbor methods. The main optimality result is that for all of these methods \(L_ n(\hat k)/L_ n(\tilde k)\to 1\) in probability, where \[ L_ n(k)=\int(\hat m_ k(x)- m(x))^ 2 dF(x). \] Further results of this kind and a numerical example are also given.

62G07 Density estimation
62J02 General nonlinear regression
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