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Nonparametric regression function estimation using interaction least squares splines and complexity regularization. (English) Zbl 1093.62528
Summary: Let $$(X,Y)$$ be a pair of random variables with $$\text{supp}(X)\subseteq [0,1]^l$$ and $$EY^2<\infty$$, $$l\geq 1$$. Let $$m^*$$ be the best approximation of the regression function of $$(X, Y)$$ by sums of functions of at most $$d$$ variables ($$1\leq d\leq l)$$. Estimation of $$m^*$$ from i.i.d. data is considered.
For the estimation interaction least squares splines, which are defined as sums of polynomial tensor product splines of at most d variables, are used. The knot sequences of the tensor product splines are chosen equidistant. Complexity regularization is used to choose the number of the knots and the degree of the splines automatically using only the given data.
Without any additional condition on the contribution of $$(X, Y)$$ the weak and strong $$L^2$$-consistency of the estimate is shown. Furthermore, for even $$p \geq 1$$ and every distribution of $$(X, Y)$$ with $$\text{supp}(X)\subseteq [0,1]^l$$, $$Y$$ bounded and $$m^*$$ $$p$$-smooth, the integrated squared error of the estimate achieves up to a logarithmic factor the (optimal) rate $$n[(2p)/(2p+d)]$$.

##### MSC:
 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference
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