Asymptotics for $L_{1}$-wavelet method for nonparametric regression

Abstract

Wavelets are particularly useful because of their natural adaptive ability to characterize data with intrinsically local properties. When the data contain outliers or come from a population with a heavy-tailed distribution, L 1-estimation should obtain a better fit. In this paper, we propose a L 1-wavelet method for nonparametric regression, and derive the asymptotic properties of the L 1-wavelet estimator, including the Bahadur representation, the rate of convergence and asymptotic normality. The rate of convergence of it is comparable with the optimal convergence rate of the nonparametric estimation in nonparametric models, and it does not require the continuously differentiable conditions of a nonparametric function.

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

“…Proof. First of all, we note that the normalized kernel K h satisfies the Lipschitz condition with constant k2 2k+1 L k h −k−1 , and for the sets N h (t) defined in (19), under the conditions of the lemma, we have the estimate…”

“…In the case of deterministic design in the vast majority of works, one or another conditions for the design regularity are assumed (for example, see [10][11][12][13][14][15][16][17][18][19]). In papers dealing with random design, independent identically distributed design elements are often considered [1,11,[20][21][22][23][24][25][26][27][28][29][30].…”

Universal Local Linear Kernel Estimators in Nonparametric Regression

“…In the case of fixed design, the vast majority of papers make certain assumptions on the regularity of design (see Zhou and Zhu, 2020;Benelmadani et al, 2020;Tang et al, 2018;Gu et al, 2007;Benhenni et al, 2010;Müller and Prewitt, 1993;Ahmad and Lin, 1984;Georgiev, 1988Georgiev, , 1990. In univariate models, the nonrandom design points X i are most often restricted by the formula X i = g(i/n) + o(1/n) with a function g of bounded variation, where the error term o(1/n) is uniform in i = 1, .…”

“…In the recent papers [23][24][25][26], nonstationary sequences of design elements with one or another special type of dependence are considered (Markov chains, autoregression, partial sums of moving averages, etc.). In the case of fixed design, in the overwhelming majority of works, certain conditions for the regularity of the design are assumed (e.g., see [9,10,[27][28][29][30][31][32][33]). So, the nonrandom design points z i are most often given by the formula z i = g(i/n) + o(1/n) with some function g of bounded variation, where the error o(1/n) is uniform in all i = 1, .…”

Universal local linear kernel estimators in nonparametric regression

“…In the recent papers [8], [28], [42], and [55], nonstationary sequences of design elements with one or another special type of dependence are considered (Markov chains, autoregression, partial sums of moving averages, etc.). In the case of fixed design, in the overwhelming majority of works, certain conditions for the regularity of the design are assumed (e.g., see [1]- [3], [17], [20], [21], [54], [57], [65]). So, the nonrandom design points z i are most often given by the formula z i = g(i/n) + o(1/n) with some function g of bounded variation, where the error o(1/n) is uniform in all i = 1, .…”