Abstract
In this paper, we establish the strong consistency and complete consistency of the Priestley–Chao estimator in nonparametric regression model with widely orthant dependent errors under some general conditions. We also obtain the rates of strong consistency and complete consistency. We show that the rates can approximate to O(n^{-1/3}) under appropriate conditions. The results obtained in the paper improve or extend the corresponding ones to widely orthant dependent assumptions.
Highlights
Consider the following nonparametric regression model: Yi = f (xi) + εi, 1 ≤ i ≤ n, (1)where f is an unknown function defined in the interval [0, 1], {Y1, . . . , Yn} are n observations at the fixed points {x1, . . . , xn}, and {εi, 1 ≤ i ≤ n} are random errors
Priestley and Chao [1] established the weak consistency of the estimator with i.i.d. errors; Benedetti [2] further studied the strong convergence and asymptotic normality with i.i.d. errors; Yang and Wang [3] obtained the strong consistency and
We further study the consistency properties of estimator (2) under a much more general dependent structure, that is, a widely orthant dependent structure
Summary
Wang et al [10] obtained uniform asymptotic estimates of finite-time ruin probability with WOD claim sizes; Wang and Cheng [11] investigated the basic renewal theorems and complete convergence for random walks with WOD increments; Liu et al [12] and Chen et al [13] improved and extended the preceding results; Wang et al [14] gave a result on uniform asymptotics of the finite-time ruin probability for all times with WLOD interoccurrence times; Shen [15] obtained a Bernstein-type inequality for WOD random variables; Qiu and Chen [16] obtained the complete convergence and complete moment convergence for weighted sums of WOD random variables under some general conditions; Wang et al [17] studied the complete convergence for arrays of WOD random variables and gave its application to complete consistency of the weighted estimator in a nonparametric regression model based on WOD errors; Yang et al [18] presented the Bahadur representation of sample quantiles for WOD random variables; Wang and Hu [19] established the weak consistency, strong consistency, complete consistency, and their convergence rates for the nearest-neighbor estimator of the density function based on WOD samples; Wu et al [20] investigated the complete moment convergence for arrays of WOD random variables; and so on. Remark 2.2 Yang and Wang [3] obtained the results of strong consistency for the estimator fn(x) with NA errors under the assumption δn/hn = O(n–1/r(log n)–1–ρ) for some ρ > 1.
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