Abstract
In the present paper, we consider the nonparametric regression model with random design based on (X-t,Y-t)(t >= 0) a R(d)xR(q)-valued strictly stationary and ergodic continuous time process, where the regression function is given bym(x,psi)=E(psi(Y) divide X=x)), for a measurable function psi:R-q -> R. We focus on the estimation of the location Theta (mode) of a unique maximum of m(center dot,psi)by the location Theta(M) over capT of a maximum of the Nadaraya-Watson kernel estimatorm T(center dot,psi)for the curvem(center dot,psi). Within this context, we obtain the consistency with rate and the asymptotic normality results for Theta(M) over cap Tunder mild local smoothness assumptions on m(center dot,psi) and the design density f(center dot) of X. Beyond ergodicity, any other assumption is imposed on the data. This paper extends the scope of some previous results established under the mixing condition. The usefulness of our results will be illustrated in the construction of confidence regions.