Abstract
The Nadaraya-Watson estimator is certainly the most popular nonparametric regression estimator. The asymptotic bias and variance of this estimator, say \(\hat{m}(x)\), are well known. Nevertheless, its higher moments are rarely mentioned in the literature. In this paper, explicit formulas for asymptotic higher moments, such as \(E((\hat{m}(x)-m(x))^{\gamma})\) or \(E((\hat{m}(x)-E(\hat{m}(x)))^{\gamma} )\), for γ any positive integer, are derived and illustrated by some examples. In particular, explicit asymptotic expressions for the L γ-errors of \(\hat{m}(x)\), for any γ, are shown. These results also allow one to give alternative proofs for the asymptotic normality and a Large Deviation Principle for the estimator. Other kernel regression estimators are also briefly discussed.
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References
Bierens, H.J. (1987). Kernel estimators of regression functions. In Advances in Econometrics, (T. F. Bewley, ed.), Cambridge University Press, 99–144.
Bucklew, J.A. (2004). Introduction to rare event simulation. Springer, New York.
Cao-Abad, R. (1991). Rate of convergence for the wild bootstrap in nonparametric regression. Ann. Statist., 19, 2226–2231.
Dembo, A. and Zeitouni, O. (1998). Large deviations techniques and applications, (2nd ed.). Springer-Verlag, New York.
Doukhan, P. and Lang, G. (2009). Evaluation for moments of a ratio with application to regression estimation. Bernoulli, 15, 1259–1286.
Fan, J. and Gijbels, I. (1996). Local polynomial modelling and its applications. Chapman and Hall, London.
Hall, P. (1992a). The bootstrap and Edgeworth expansion. Springer-Verlag, New-York.
Hall, P. (1992b). On the removal of skewness by transformation. J. R. Statist. Soc. B, 54, 221–228.
Härdle, W. and Mammen, E. (1991). Bootstrap methods in nonparametric regression. In Nonparametric Functional Estimation and Related Topics, (G. Roussas, ed.), Kluwer Academic Publisher, 111–123.
Härdle, W., Müller, M., Sperlich, S. and Werwatz, A. (2004). Nonparametric and semiparametric models. An introduction. Springer-Verlag, New-York.
Louani, D. (1999). Some large deviations limit theorems in conditional nonparametric statistics. Statistics, 33, 171–196.
Nadaraya, E.A. (1964). On estimating regression. Theory Probab. Applic., 9, 141–142.
Wand, M.P. (1990). On exact L 1 rates in nomparametric kernel regression. Scand. J. Statist., 251–256.
Wand, M.P. and Jones, M.C. (1995). Kernel smoothing. Chapman and Hall, London.
Watson, G.S. (1964). Smooth regression analysis. Sankhya A, 26, 359–372.
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Geenens, G. Explicit Formula for Asymptotic Higher Moments of the Nadaraya-Watson Estimator. Sankhya A 76, 77–100 (2014). https://doi.org/10.1007/s13171-013-0035-y
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DOI: https://doi.org/10.1007/s13171-013-0035-y