Universal Local Linear Kernel Estimators in Nonparametric Regression

Abstract

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.

Multivariate Universal Local Linear Kernel Estimators in Nonparametric Regression: Uniform Consistency

“…In the present paper, we continue to develop the concept of dense data proposed in [51][52][53]. In these papers, it is established that to restore the regression function, it is enough to know the noisy values of this function on some dense (in one sense or another) set of points from the regression function domain.…”

“…Previously, similar ideas were implemented in [51,52] for local constant estimators and in [53] for local linear estimators with a univariate design. The estimators from [53] are a particular case (for k = 1) of the estimators proposed here.…”

“…Previously, similar ideas were implemented in [51,52] for local constant estimators and in [53] for local linear estimators with a univariate design. The estimators from [53] are a particular case (for k = 1) of the estimators proposed here. Note that the construction of estimators from [53], when the weights of the weighted least squares method are the spacing statistics constructed by the variational series of design points, has no direct generalization to the case of functions of many variables.…”

Adaptive edge detection of rebar thread head image based on improved Canny operator

“…where w r (r ∈ (−N max , N max ), i = 1, … , M , j = 1, … , N ) represents an adaptive filtering window, f (i, j ) is the filtered pixel value, and the weighting coefficient w(i, j ) can be described as Equation (16).…”

“…Many scholars at home and abroad had made various improvements to the structure and methods of the Canny operator to improve the accuracy and universality of edge detection. In terms of filtering and denoising, multiple filters were proposed to replace Gaussian filters to smooth and denoise images to improve the signal-to-noise ratio of images, such as local linear kernel smoothing filters, statistical filters, wavelet transforms, morphological filters, adaptive median filters, fast guidance filters and so on [16][17][18][19][20][21][22][23][24], which had effectively suppressed noise and achieved good results. In addition, a dual filter combining adaptive fuzzy median filtering and bilateral filtering was proposed to eliminate salt and pepper noise in images [25].…”