Estimation of a projection-pursuit type regression model. (English) Zbl 0736.62055

A nonparametric estimate of the conditional mean value \(m_ 0(X)=E(Y\mid X=x)\) is proposed for \(X\) being assumed \(d\)-dimensional. The conditional mean value is assumed to have the form \[ m_ 0(X)=\mu_ 0+\sum^{K_ 0}_{j=1}\theta_ j(\beta^ T_ jX), (1) \] where \(\theta_ j\) are \(q\)-times continuously differentiable, bounded real functions with the \(q\)-th derivative being Lipschitz and \(\text{ang}(\{\beta_ 1,\ldots,\beta_{K_ 0}\})\geq M_ 0>0\), where \(\text{ang}(\{\beta_ 1,\ldots,\beta_{K_ 0})\) denotes the minimum among all angles between \(\beta_ i\) and the linear space spanned by \(\{\beta_ 1,\ldots,\beta_{K_ 0}\}\backslash\{\beta_ i\}\) for \(i=1,\ldots,K_ 0\) (for \(K_ 0=1\) it is defined as \(\pi/2\)). The estimator is considered in the form (\(1\leq k\leq d\)) \(m(x)=\mu+\sum^ k_{j=1}s_ j(\alpha_ jx)\), where \(\mu\) is a constant and each \(s_ j\) is a polynomial spline of degree \(q\) on \([-1,1]\) with equispaced knots of distance \(2/N\) and \(\text{ang}(\{\alpha_ 1,\ldots,\alpha_ k\})\geq M>0\). The density of \((Y,X)\) is assumed to be such that:
i) The marginal density of \(X\) is bounded away from zero and infinity on a compact set containing the unit ball \(C\) in \(R^ r\);
ii) \(\inf_ x\hbox{ var}(Y\mid X=x)>0.\)
On the estimator \(m(x)\) the following constraints are imposed:
i) There exists a positive integer \(\tau>(2d+5)(2p+1)/(2\gamma-1)\) (where \(p\in(q,q+1]\) and \(\gamma\in(1/2,1))\), and a positive constant \(c_ 3\) such that \[ \sup_ x E[| Y-m(x)|^{4\tau}\mid X=x]\leq c_ 3; \]
ii) \(M\leq M_ 0\).
Under these constraints the estimator is defined as the least squares estimator but only observations falling into the unit ball \(C\) are assumed, i.e. \[ \hat m_ n(x)=\arg\min\left\{\sum^ n_{i=1} [y_ i- m(x_ i)]^ 21_ C(x_ i)\right\}. \] The main result of the paper is then: \[ \lim_{n\rightarrow\infty}\sup_{\theta\in\Theta_{p,d}} P_ \theta \left\{n^{-1}\sum^ n_{i=1} [\hat m_ n(x_ i)-m_ 0(x_ i)]^ 2 1_ C(x_ i)\geq cn^{-2p/(2p+1)}\right\}=0, \] where \(\Theta_{p,d}\) denotes the collection of probability measures such that \(E(Y\mid X=x)\) has the form (1).
As follows from this result, the imposed constraints imply that the rate of convergence of the estimators does not depend on the dimension \(d\).


62J02 General nonlinear regression
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62H99 Multivariate analysis
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