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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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