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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 $$d$$.

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