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The use of polynomial splines and their tensor products in multivariate function estimation. (With discussion). (English) Zbl 0827.62038
Summary: Let $$X_1, \dots, X_M$$, $$Y_1, \dots, Y_N$$ be random variables, and set $${\mathbf X}= (X_1, \dots, X_M)$$ and $${\mathbf Y}= (Y_1, \dots$$, $$Y_N)$$. Let $$\varphi$$ be the regression or logistic or Poisson regression function of $${\mathbf Y}$$ on $${\mathbf X}$$ $$(N=1)$$ or the logarithm of the density function of $${\mathbf Y}$$ or the conditional density function of $${\mathbf Y}$$ and $${\mathbf X}$$. Consider the approximation $$\varphi^*$$ to $$\varphi$$ having a suitably defined form involving a specified sum of functions of at most $$d$$ of the variables $$x_1, \dots, x_M$$, $$y_1, \dots, y_N$$ and, subject to this form, selected to minimize the mean squared error of approximation or to maximize the expected log-likelihood or conditional log-likelihood, as appropriate, given the choice of $$\varphi$$. Let $$p$$ be a suitably defined lower bound to the smoothness of the components of $$\varphi^*$$. Consider a random sample of size $$n$$ from the joint distribution of $${\mathbf X}$$ and $${\mathbf Y}$$.
Under suitable conditions, the least squares or maximum likelihood method is applied to a model involving nonadaptively selected sums of tensor products of polynomial splines to construct estimates of $$\varphi^*$$ and its components having the $$L_2$$ rate of convergence $$n^{-p/ (2p+ d)}$$.

##### MSC:
 62G07 Density estimation 62J12 Generalized linear models (logistic models) 62H12 Estimation in multivariate analysis 62G20 Asymptotic properties of nonparametric inference
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