Geodesic
Asymptotic bounds on the quality of the nonparametric regression estimation in~$\mathscr L_p$
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VI, Tome 97 (1980), pp. 88-101

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We consider the asymptotic set-up of the following nonparametric problem. Choose the points of measurement $X_1,\dots,X_N$ and estimate the unknown function $f$ on the base of observations $$ Y_i=f(X_i)+G_i(X_i,\omega),\quad i=1,\dots,N, $$ where noise variables $G_1,\dots,G_N$ are independent when $X_1,\dots,X_N$ are fixed. We suppose that the deviation of estimator from regression function $f$ is measured in $\mathscr L_p$ metrix, $1\le p\infty$. The case $p=\infty$ we consider in [1]. 
@article{ZNSL_1980_97_a9,
     author = {I. A. Ibragimov and R. Z. Khas'minskii},
     title = {Asymptotic bounds on the quality of the nonparametric regression estimation in~$\mathscr L_p$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {88--101},
     publisher = {mathdoc},
     volume = {97},
     year = {1980},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1980_97_a9/}
}
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I. A. Ibragimov; R. Z. Khas'minskii. Asymptotic bounds on the quality of the nonparametric regression estimation in~$\mathscr L_p$. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VI, Tome 97 (1980), pp. 88-101. http://geodesic.mathdoc.fr/item/ZNSL_1980_97_a9/