The asymptotic normality of the linear weighted estimator in nonparametric regression models

References

• Chen, P. Y., and S. X. Gan. 2008. On moments of the maximum of normed partial sums of ρ-mixing random variables. Statistics and Probability Letters 78:1215–21.
   Google Scholar
• Ding, L. W., and Y. M. Li. 2016. The Berry-Esseen bounds of wavelet estimator for regression model whose errors form a linear process with a ρ-mixing. Journal of Inequalities and Applications 2016, Article ID 107, 12.
   Google Scholar
• Dobrushin, R. L. 1956. The central limit theorem for non-stationary Markov chain. Theory of Probability and Its Applications 1:72–88.
   Google Scholar
• Georgiev, A. A. 1988. Consistent nonparametric multiple regression: the fixed design case. Journal of Multivariate Analysis 25 (1):100–10.
   Web of Science ®Google Scholar
• Georgiev, A. A., and W. Greblicki. 1986. Nonparametric function recovering from noisy observations. Journal of Statistical Planning and Inference 13:1–14.
   Web of Science ®Google Scholar
• Hu, S. H., C. H. Zhu, Y. B. Chen, and L. C. Wang. 2002. Fixed-design regression for linear time series. Acta Mathematica Scientia 22B (1):9–18.
   Google Scholar
• Ibragimov, L. A. 1962. Some limit theorems for stationary process. Theory of Probability and its Applications 7:349–82.
   Google Scholar
• Kolmogorov, A. N., and Y. A. Rozanov. 1960. On strong mixing conditions for stationary Gaussian processes. Theory of Probability and its Applications 5:204–08.
   Google Scholar
• Li, Y. M., J. H. Guo, and N. Y. Li. 2012. Some inequalities for a LNQD sequence with applications. Journal of Inequalities and Applications Vol. 2012, Article ID 216, 9.
   Google Scholar
• Liang, H. Y., and B. Y. Jing. 2005. Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences. Journal of Multivariate Analysis 95:227–45.
   Web of Science ®Google Scholar
• Müller, H. G. 1987. Weak and universal consistency of moving weighted averages. Periodica Mathematica Hungarica 18 (3):241–50.
   Google Scholar
• Peligrad, M. 1985. An invariance principle for ϕ-mixing sequences. The Annals of Probability 13:1304–13.
   Web of Science ®Google Scholar
• Roussas, G. G., L. T. Tran, and D. A. Ioannides. 1992. Fixed design regression for time series: asymptotic normality. Journal of Multivariate Analysis 40:262–91.
   Web of Science ®Google Scholar
• Sen, P. K. 1974. Weak convergence of multidimensional empirical processes for stationary ϕ-mixing processes. The Annals of Probability 2:147–54.
   Web of Science ®Google Scholar
• Shao, Q. M. 1995. Maximal inequalities for partial sums of ρ-mixing sequences. The Annals of Probability 23 (2):948–65.
   Web of Science ®Google Scholar
• Shen, A. T., Y. Shi, W. J. Wang, and B. Han. 2012. Bernstein-type inequality for weakly dependent sequence and its applications. Revista Matemática Complutense 25:97–108.
   Web of Science ®Google Scholar
• Shen, A. T. 2013. Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models. Abstract and Applied Analysis Volume 2013, Article ID 862602, 9 pages.
   Google Scholar
• Shen, A. T., X. H. Wang, and J. M. Ling. 2014. On complete convergence for non-stationary ϕ-mixing random variables. Communications in Statistics-Theory and Methods 43 (22):4856–66.
   Web of Science ®Google Scholar
• Shen, A. T. 2016. Complete convergence for weighted sums of END random variables and its application to nonparametric regression models. Journal of Nonparametric Statistics 28 (4):702–15.
   Web of Science ®Google Scholar
• Stone, C. J. 1977. Consistent nonparametric regression. The Annals of Statistics 5 (4):595–620.
   Web of Science ®Google Scholar
• Tran, L. T., G. G. Roussas, S. Yakowitz, and B. T. Van. 1996. Fixed-design regression for linear time series. The Annals of Statistics 24:975–91.
   Web of Science ®Google Scholar
• Utev, S. A. 1990. On the central limit theorem for ϕ-mixing arrays of random variables. Theory of Probability and its Applications 35 (1):131–9.
   Web of Science ®Google Scholar
• Wang, X. J., and S. H. Hu. 2012. Some Baum-Katz type results for ϕ-mixing random variables with different distributions. RACSAM 106 (2):321–31.
   Web of Science ®Google Scholar
• Wang, X. J., X. Deng, L. L. Zheng, and S. H. Hu. 2014a. Complete convergence for arrays of rowwise negatively superadditive dependent random variables and its applications. Statistics: A Journal of Theoretical and Applied Statistics 48 (4):834–50.
   Google Scholar
• Wang, X. J., C. Xu, T. C. Hu, A. Volodin, and S. H. Hu, 2014b. On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST 23(3):607C–29.
   Google Scholar
• Wang, X. J., L. L. Zheng, C. Xu, and S. H. Hu. 2015. Complete consistency for the estimator of nonparametric regression models based on extended negatively dependent errors. Statistics: A Journal of Theoretical and Applied Statistics 49 (2):396–407.
   Google Scholar
• Wu, Q. Y. 2006. Probability Limit Theory for Mixing Sequences. Beijing: Science Press of China.
   Google Scholar
• Wu, Q. Y., and Y. Y. Jiang. 2010. Chover-type laws of the k-iterated logarithm for ρ˜-mixing sequences of random variables. Journal of Mathematical Analysis and Applications 366:435–43.
   Web of Science ®Google Scholar
• Wu, Y., X. J. Wang, and S. H. Hu. 2017. Complete moment convergence for weighted sums of weakly dependent random variables and its application in nonparametric regression model. Statistics and Probability Letters 127:56–66.
   Web of Science ®Google Scholar
• Yang, S. C. 2007. Maximal moment inequality for partial sum of strong mixing sequences and application. Acta Mathematica Sinica, English Series 23 (6):1013–24.
   Web of Science ®Google Scholar
• Yang, W. Z., X. J. Wang, X. H. Wang, and S. H. Hu. 2012. The consistency for estimator of nonparametric regression model based on NOD errors. Journal of Inequalities and Applications, Volume 2012, Article ID 140, 13 pages.
   Google Scholar
• Zheng, L. L., Y. Ding, and X. J. Wang. 2017. Asymptotic normality for the estimator of non parametric regression model under ϕ-mixing errors. Communications in Statistics-Theory and Methods 46 (14):6764–73.
   Web of Science ®Google Scholar
• Zhou, X. C., J. G. Lin, and C. M. Yin. 2013. Asymptotic properties of wavelet-based estimator in nonparametric regression model with weakly dependent processes. Journal of Inequalities and Applications, Volume 2013, Article ID 261, 18 pages.
   Google Scholar