Asymptotic normality of the Nadaraya-Watson estimator for non-stationary functional data and applications to telecommunications. L. ASPIROT ∗ , K. BERTIN † , G. PERERA ∗ † Departamento de Estad´ ıstica, CIMFAV, Universidad de Valpara´ ıso, Chile ∗ Instituto de Matem´atica y Estad´ ıstica “Rafael Laguardia”, Facultad de Ingenier´ ıa, Uruguay. July 6, 2007 Abstract We study a non-parametric regression model where the explanatory variable is a non- stationary dependent functional data and the response variable is scalar. Supposing that the explanatory variable is a non-stationary mixture of stationary processes and general conditions of dependence of the observations (implied in particular by weakly dependence), we obtain the asymptotic normality of the Nadaraya-Watson estimator. Under some ad- ditional regularity assumptions on the regression function, it can be obtained asymptotic confidence intervals for the regression function. We apply this result to the estimation of the quality of service on the Internet where the cross traffic is non-stationary. Keywords: Non-parametric regression, functional data, asymptotic normality, non-stationarity, dependence, quality of service. AMS subject classification: 62G05, 62G08, 62G20. 1 Introduction We study the problem of estimating a non-parametric regression function φ when the explana- tory variable is functional and may be non-stationary and dependent. We observe {X,Y } = {(X i ,Y i ): i =0,...,n − 1} such that Y i = φ(X i )+ ε i ,i =0,...,n − 1, (1) where φ is a function φ : D→ R, X i ∈D, Y i ∈ R, the ε i ’s are centered and independent of the X i ’s and D is a semi-normed linear space with semi-norm ‖·‖. * Instituto de Matem´atica y Estad´ ıstica “Rafael Laguardia”, (IMERL), Facultad de Ingenier´ ıa, Julio Herrera y Reissig 565, Montevideo 11200, Tel-fax: (5982)7110621, Emails: aspirot@fing.edu.uy, pereragonzalo@yahoo.fr † Departamento de Estad´ ıstica, CIMFAV, Universidad de Valpara´ ıso, Gran Breta˜ na 1091, 4to piso, Playa Ancha, Valpara´ ıso, Chile, Tel: (56)322508324, Email: karine.bertin@uv.cl 1