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Rates of convergence for the estimates of the optimal transformations of variables. (English) Zbl 0733.62054

Consider a \((d+1)\)-dimensional random vector. L. Beiman and J. H. Friedman [J. Am. Stat. Assoc. 80, 580-619 (1985; Zbl 0594.62044)] considered transformations h(Y), \(\phi_ 1(X_ 1),...,\phi_ d(X_ d)\) so that the transformed random variables have zero means, E \(h^ 2(Y)=1\), E \(\phi\) \({}^ 2_ j(X_ j)<\infty\) for \(j=1,...,d\). Let \[ {\tilde \phi}(X)=\sum^{d}_{j=1}\phi_ j(X_ j)\text{ and } E^ 2(h,{\tilde \phi})=E(h(Y)-\phi (X))^ 2, \] (h\({}^*,{\tilde \phi}^*)\) are called optimal if they minimize \(E^ 2(h,{\tilde \phi})\). Breiman and Friedman proved that optimal transformations exist, but they are not unique, in general.
The authors consider spline approximations to h, \(\phi_ 1,...,\phi_ d\), and construct estimators of these splines via a sample of size n. They prove results on the rate of convergence of the constructed estimators to the optimal transformations. These rates are comparable with those in the usual nonparametric estimation of a density.

MSC:

62G20 Asymptotic properties of nonparametric inference
62G07 Density estimation

Citations:

Zbl 0594.62044
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