Asymptotic normality of kernel estimators of the conditional mode under strong mixing hypothesis

Abstract

Let (X _n ,Y _n ) _n ≤1 be a R ^d ×R valued stationary process. Define the estimator of the conditional mode of Y ₁ given X ₁=x as the random variable θ _n (x) that maximizes a kernel estimator of the conditional density of Y ₁ given X ₁ = x. We establish asymptotic normality of θ _n (x) when the process (X _n ,Y _n ) _n ≤1 is assumed to be strongly mixing. We derive from our results asymptotic normality of a predictor and propose a confidence bands for the conditional mode function. A simulation study shows how good the normality of the conditional mode function estimator is when dealing with samples of finite sizes.