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In this paper we prove large deviations principles for the averagedstochastic approximation method for the estimation of a regression functionintroduced by A. Mokkadem et al. [Revisiting R\'ev\'esz's stochasticapproximation method for the estimation of a regression function, ALEA Lat.Amm. J. Probab. Math. Stat. 6 (2009), 63-114]. We show that the averagedstochastic approximation algorithm constructed using the weight sequence whichminimize the asymptotic variance gives the same pointwise LDP as theNadaraya-Watson kernel estimator. Moreover, we give a moderate deviationsprinciple for these estimators. It turns out that the rate function obtained inthe moderate d\'eviations principle for the averaged stochastic approximationalgorithm constructed using the weight sequence which minimize the asymptoticvariance is larger than the one obtained for the Nadaraya-Watson estimator andthe one obtained for the semi-recursive estimator.

Large and moderate deviation principles for averaged stochastic approximation method for the estimation of a regression function