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conditional mode, conditional quantile, conditional cumulative distribution, derivatives of conditional density, nonparametric estimation, kernel estimator, functional random variable, small ball probability, gaussian processes, diffusion processes, chemiometrics,

We consider kernel regression estimate when both the response variable and the explanatory one are functional. The rates of uniform almost complete convergence are stated as function of the small ball probability of the predictor and as function of the entropy of the set on which uniformity is obtained.

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-Saïd. A generalized -approach for a kernel estimator of conditional quantile with functional regressors: Consistency and asymptotic normality. Statistics and Probability Letters, Elsevier, 2009, 79 (8) This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPTA generalized L Abstract: A kernel estimator of the conditional quantile is defined for a scalar response variable given a covariate taking values in a semi-metric space. The approach generalizes the median's L 1 -norm estimator.The almost complete consistency and asymptotic normality are stated.Keywords Asymptotic distribution · functional data · kernel estimate · robust estimation · small balls probability.

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a b s t r a c tWe propose a family of robust nonparametric estimators for a regression function based on the kernel method. We establish the asymptotic normality of the estimator under the concentration property on small balls probability measure of the functional explanatory variable when the observations exhibit some kind of dependence. This approach can be applied in time series analysis to make prediction and build confidence bands. We illustrate our methodology on the US electricity consumption data.