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Estimating the error distribution in nonparametric multiple regression with applications to model testing. (English) Zbl 1185.62078
Summary: We consider the estimation of the error distribution in a heteroscedastic nonparametric regression model with multivariate covariates. As estimator we consider the empirical distribution function of the residuals, which are obtained from multivariate local polynomial fits of the regression and variance functions, respectively. Weak convergence of the empirical residual process to a Gaussian process is proved. We also consider various applications for testing model assumptions in nonparametric multiple regression. The model tests obtained are able to detect local alternatives that converge to zero at an $$n^{-1/2}$$-rate, independent of the covariate dimension. We consider in detail a test for additivity of the regression function.

##### MSC:
 62G08 Nonparametric regression and quantile regression 62G10 Nonparametric hypothesis testing 62G20 Asymptotic properties of nonparametric inference 62G30 Order statistics; empirical distribution functions 60F05 Central limit and other weak theorems
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##### References:
 [1] Akritas, M.G.; Van Keilegom, I., Nonparametric estimation of the residual distribution, Scand. J. statist., 28, 549-567, (2001) · Zbl 0980.62027 [2] Müller, U.U.; Schick, A.; Wefelmeyer, W., Estimating the error distribution function in semiparametric regression, Statist. decisions, 25, 1-18, (2007) · Zbl 1137.62023 [3] Van Keilegom, I.; González-Manteiga, W.; Sánchez Sellero, C., Goodness-of-fit tests in parametric regression based on the estimation of the error distribution, Test, 17, 401-415, (2008) · Zbl 1196.62049 [4] Dette, H.; Neumeyer, N.; Van Keilegom, I., A new test for the parametric form of the variance function in non-parametric regression, J. roy. statist. soc. ser. B, 69, 903-917, (2007) [5] Pardo-Fernández, J.C.; Van Keilegom, I.; González-Manteiga, W., Comparison of regression curves based on the estimation of the error distribution, Statistica sinica, 17, 1115-1137, (2007) · Zbl 1133.62031 [6] Pardo-Fernández, J.C., Comparison of error distributions in nonparametric regression, Statist. probab. lett., 77, 350-356, (2007) · Zbl 1106.62044 [7] Neumeyer, N.; Dette, H.; Nagel, E.-R., Bootstrap tests for the error distribution in linear and nonparametric regression models, Austr. New Zealand J. statist., 48, 129-156, (2005) · Zbl 1108.62032 [8] Fan, J.; Gijbels, I., Local polynomial modelling and its applications, (1996), Chapman & Hall London · Zbl 0873.62037 [9] Ruppert, D.; Wand, M.P., Multivariate locally weighted least squares regression, Ann. statist., 22, 1346-1370, (1994) · Zbl 0821.62020 [10] Härdle, W.; Tsybakov, A., Local polynomial estimators of the volatility function in nonparametric autoregression, J. econometrics, 81, 223-242, (1997) · Zbl 0904.62047 [11] J.M. Rodríguez-Póo, S. Sperlich, P. Vieu, An adaptive specification test for semiparametric models, (submitted for publication) (paper available at http://papers.ssrn.com/sol3/) 2005. [12] Chen, S.X.; Van Keilegom, I., A goodness-of-fit test for parametric and semiparametric models in multiresponse regression, Bernoulli, 15, 955-976, (2009) · Zbl 1200.62047 [13] Gozalo, P.L.; Linton, O.B., A nonparametric test of additivity in generalized nonparametric regression with estimated parameters, J. econometrics, 104, 1-48, (2001) · Zbl 0978.62032 [14] Dette, H.; von Lieres und Wilkau, C., Testing additivity by kernel based methods — what is a reasonable test?, Bernoulli, 7, 669-697, (2001) · Zbl 1005.62037 [15] Yang, L.; Park, B.U.; Xue, L.; Härdle, W., Estimation and testing for varying coefficients in additive models with marginal integration, J. amer. statist. assoc., 101, 1212-1227, (2006) · Zbl 1120.62317 [16] Newey, W.K., Kernel estimation of partial means, Econometric theory, 10, 233-253, (1994) [17] Tjøstheim, D.; Auestad, B.H., Nonparametric identification of nonlinear time series: projections, J. amer. statist. assoc., 89, 1398-1409, (1994) · Zbl 0813.62036 [18] Linton, O.B.; Nielsen, J.P., A kernel method of estimating structured nonparametric regression based on marginal integration, Biometrika, 82, 93-101, (1995) · Zbl 0823.62036 [19] Neumeyer, N., Smooth residual bootstrap for empirical processes of nonparametric regression residuals, Scand. J. statist., 36, 204-228, (2009) · Zbl 1194.62051 [20] R Development Core Team, R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org. 2009. [21] Neumeyer, N.; Sperlich, S., Comparison of separable components in different samples, Scand. J. statist., 33, 477-501, (2006) · Zbl 1114.62054 [22] Van Keilegom, I.; Akritas, M.G., Transfer of tail information in censored regression models, Ann. statist., 27, 1745-1784, (1999) · Zbl 0957.62034 [23] Einmahl, J.; Van Keilegom, I., Specification tests in nonparametric regression, J. econometrics, 143, 88-102, (2008) · Zbl 1418.62156 [24] Dette, H.; Pardo-Fernández, J.C.; Van Keilegom, I., Goodness-of-fit tests for multiplicative models with dependent data, Scand. J. statist., 36, 782-799, (2009) · Zbl 1224.62008 [25] Masry, E., Multivariate local polynomial regression for time series: uniform strong consistency and rates, J. time series anal., 17, 571-599, (1996) · Zbl 0876.62075 [26] J.L. Ojeda, Hölder continuity properties of the local polynomial estimator 2008 (submitted for publication) (paper available at http://www.unizar.es/galdeano/preprints/lista.html). [27] van der Vaart, A.W.; Wellner, J.A., Weak convergence and empirical processes, (1996), Springer New York · Zbl 0862.60002
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