New local linear estimators are proposed for a wide class of nonparametric regression models. The estimators are uniformly consistent regardless of satisfying traditional conditions of dependence of design elements. The estimators are the solutions of a specially weighted least-squares method. The design can be fixed or random and does not need to meet classical regularity or independence conditions. As an application, several estimators are constructed for the mean of dense functional data. The theoretical results of the study are illustrated by simulations. An example of processing real medical data from the epidemiological cross-sectional study ESSE-RF is included. We compare the new estimators with the estimators best known for such studies.

We show that Nadaraya–Watson estimators in nonparametric regression may be uniformly consistent without any specification of the design correlation structure. In contrast to the predecessors’ results, the design is not required to be fixed or consisted of independent or weakly dependent random variables under various dependence conditions, and the design random observations are not necessarily identically distributed and nondegenerate. We suggest a new simple sufficient condition on the design which provides the uniform consistency of such estimators. Key words and phrases: nonparametric regression, Nadaraya–Watson estimators, uniform consistency, strongly dependent data.

We study the asymptotic behavior of one-step M-estimators based on not necessarily independent identically distributed observations. In particular, we find conditions for asymptotic normality of these estimators. Asymptotic normality of one-step M-estimators is proven under a wide spectrum of constraints on the exactness of initial estimators. We discuss the question of minimal restrictions on the exactness of initial estimators. We also discuss the asymptotic behavior of the solution to an M-equation closest to the parameter under consideration. As an application, we consider some examples of one-step approximation of quasi-likelihood estimators in nonlinear regression.