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The main goal of this paper is to study the asymptotic normality of the estimate of the Conditional Distribution Function of a scalar response variable Y given a hilbertian random variable X when the observations are the quasi-associated framework. Our approach is based on the Doob’s technique. It is shown that, under the concentration property on small balls of the probability measure of the functional estimator and some regularity conditions, the kernel estimate of the three parameters (conditional density, conditional distribution and conditional hazard) are asymptotically normally distributed. The result can be applied in the asymptotic normality of the estimate of the conditional hazard Function.
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