Robust kernel regression function with uncertain scale parameter for high dimensional ergodic data using $k$-nearest neighbor estimation

In this paper, we consider a new method dealing with the problem of estimating the scoring function ${\gamma }_a$, with a constant $a$, in functional space and an unknown scale parameter under a nonparametric robust regression model. Based on the $k$ Nearest Neighbors ( $k$NN) method, the primary objective is to prove the asymptotic normality aspect in the case of a stationary ergodic process of this estimator. We begin by establishing the almost certain convergence of a conditional distribution estimator. Then, we derive the almost certain convergence (with rate) of the conditional median (scale parameter estimator) and the asymptotic normality of the robust regression function, even when the scale parameter is unknown. Finally, the simulation and real-world data results reveal the consistency and superiority of our theoretical analysis in which the performance of the $k$NN estimator is comparable to that of the well-known kernel estimator, and it outperforms a nonparametric series (spline) estimator when there are irrelevant regressors.