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Dependence and the dimensionality reduction principle. (English) Zbl 1056.62054
Summary: C. Stone’s dimensionality reduction principle [Ann. Stat. 13, 689–705 (1985; Zbl 0605.62065); ibid. 10, 1040–1053 (1982; Zbl 0511.62048)] has been confirmed on several occasions for independent observations. When dependence is expressed with \(\varphi\)-mixing, a minimum distance estimate \(\widehat\theta_n\) is proposed for a smooth projection pursuit regression-type function \(\theta\in\Theta\), that is either additive or multiplicative, in the presence of or without interactions. Upper bounds on the \(L_1\)-risk and the \(L_1\)-error of \(\widehat\theta_n\) are obtained, under restrictions on the order of decay of the mixing coefficient. The bounds show explicitly the additive effect of \(\varphi\)-mixing on the error, and confirm the dimensionality reduction principle.
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
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