Vieu, Philippe Quadratic errors for nonparametric estimates under dependence. (English) Zbl 0751.62022 J. Multivariate Anal. 39, No. 2, 324-347 (1991). The author investigates the asymptotic behavior of nonparametric curve estimates under the assumption that the observed data satisfy an \(\alpha\)-mixing condition. As estimators for density, distribution, hazard, conditional density and regression functions the well-known kernel type estimators are considered. It is shown that as in the case of independence of the observations the quadratic error criteria, average square error, integrated square error and mean integrated square error are asymptotically equivalent. This equivalence holds uniformly in the smoothing parameter and with probability one.Moreover, for all considered estimators asymptotic expansions of the mean integrated square error are derived. Thus, the statements presented here are an extension of the results known in the case of independent data to that of dependent observations. Reviewer: H.Liero (Potsdam) Cited in 1 ReviewCited in 33 Documents MSC: 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference Keywords:functional estimation; alpha-mixing condition; strong mixing condition; asymptotic equivalence; nonparametric curve estimates; hazard; conditional density; regression functions; kernel type estimators; average square error; integrated square error; mean integrated square error; equivalence; asymptotic expansions; dependent observations PDFBibTeX XMLCite \textit{P. Vieu}, J. 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