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A Journal of Theoretical and Applied Statistics
Volume 56, 2022 - Issue 1
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Research Article

The Berry–Esseen-type bound for the G-M estimator in a nonparametric regression model with α-mixing errors

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Pages 97-120 | Received 14 Oct 2020, Accepted 31 Jan 2022, Published online: 04 Mar 2022

References

  • Liang HY, Li YY. A Berry-Esseen type bound of regression estimator based on linear process errors. J Korean Math Soc. 2008;45(6):1753–1767.
  • Li YM, Wei CD. Berry-Esseen bounds for wavelet estimator of regression function under strong mixing process. Acta Math Sci Ser A. 2009;29(5):1453–1463.
  • Asghari P, Fakoor V, Sarmad M. A Berry-Esseen type bound in kernel density estimation for a random left truncation model. Comm Statist Theory Methods. 2014;21(1):115–124.
  • Asghari P, Fakoor V, Sarmad M. A Berry-Esseen type bound for the kernel density estimator of length-biased data. J Sci Islam Repub Iran. 2015;26(3):256–272.
  • Liang HY, Li DL, Miao TX. Berry-Esseen type bound of conditional mode estimation under truncation and strong mixing assumptions. Comm Statist Theory Methods. 2016;45(17):5077–5097.
  • Asghari P, Fakoor V. A Berry-Esseen type bound for the kernel density estimator based on a weakly dependent and randomly left truncated data. J Inequal Appl. 2017;2017, Article ID 1, 19 pp.
  • Wang XJ, Wu Y, Hu SH. The Berry-Esseen bounds of the weighted estimator in a nonparametric regression model. Ann Inst Stat Math. 2019;71:1143–1162.
  • Wu Y, Wang XJ, Li YM, et al. Berry-Esseen type bounds of the estimators in a semiparametric model under linear process errors with α-mixing dependent innovations. Stat: J Theoret Appl Stat. 2019;53(5):943–967.
  • Kuang NH, Li Y. Berry-Esséen bounds and almost sure CLT for the quadratic variation of the sub-bifractional Brownian motion. Stat Probab Lett. 2020;18:13–17.
  • Yang XT, Yang SC. Strong consistency of integral weight regression estimator for α-mixing samples. J Math. 2019;39(6):878–888.
  • You JH, Chen M, Chen G. Asymptotic normality of some estimators in a fixed-design semiparametric regression model with linear time series errors. J Syst Sci Complex. 2004;17(4):511–522.
  • Cuevas A, Febrero M, Fraiman R. Linear functional regression: the case of fixed design and functional response. Canadian J Stat. 2002;30(2):285–300.
  • Wu JS, Chu CK. Nonparametric estimation of a regression function with dependent observations. Stoch Process Appl. 1994;50:149–160.
  • Yang SC. Maximal moment inequality for partial sums of strong mixing sequences and application. Acta Math Sin Engl Ser. 2007;23(6):1013–1024.
  • Chen J, Li DG, Lin ZY. Asymptotic expansion for nonparametric M-estimator in a nonlinear regression model with long-memory errors. J Stat Plan Inference. 2011;141:3035–3046.
  • Gasser T, Müller HG. Kernel estimation of regression functions: in smoothing techniques for curve estimation. Heidelberg: Springer-Verlag; 1979. p. 23–68.
  • Altman NS. Kernel smoothing with correlated errors. J Am Stat Assoc. 1990;85:749–759.
  • Hart JD. Kernel regression estimation with time series errors. J R Stat Soc Ser B. 1991;53:173–188.
  • Herrmann E, Gasser T, Kneip A. Choice of bandwidth for kernel regression when residuals are correlated. Biometrika. 1992;79:783–795.
  • Liebscher E. Asymptotic normality of nonparametric estimators under α-mixing condition. Stat Probab Lett. 1999;43:243–250.
  • Wieczorek B, Ziegler K. On optimal estimation of a non-smooth mode in a nonparametric regression model with α-mixing errors. J Stat Plan Inference. 2010;140:406–418.
  • Benhenni K, Rachdi M, Su YC. The effect of the regularity of the error process on the performance of kernel regression estimators. Metrika. 2013;76:765–781.
  • Rosenblatt M. A central limit theorem and a strong mixing condition. Proc Natl Acad Sci USA. 1956;42:43–47.
  • Ekström M. A general central limit theorem for strong mixing sequences. Stat Probab Lett. 2014;94:236–238.
  • Thanh LV, Yin G. Weighted sums of strongly mixing random variables with an application to nonparametric regression. Stat Probab Lett. 2015;105:195–202.
  • Dedecker J, Merlevède F. Behavior of the Wasserstein distance between the empirical and the marginal distributions of stationary α-dependent sequences. Bernoulli. 2017;23(3):2083–2127.
  • Dedecker J, Merlevède F. Almost sure invariance principle for the Kantorovich distance between the empirical and the marginal distributions of strong mixing sequences. Stat Probab Lett. 2021;171, Article ID 108991, 9 pp.
  • Tabacu L. Weak convergence of the linear rank statistics under strong mixing conditions. Stat Probab Lett. 2018;132:28–34.
  • Lin ZY. Asymptotic normality of kernel estimates of a density function under association dependence. Acta Math Sci Ser B. 2003;23(3):345–350.
  • Mokkadem A. Mixing properties of ARMA processes. Stoch Process Appl. 1988;29(2):309–315.
  • Liang HY, Fan GL. Berry-Esseen type bounds of estimators in a semiparametric model with linear process errors. J Multivar Anal. 2009;100(1):1–15.
  • Zhang JJ, Liang HY, Amei A. Asymptotic normality of estimators in heteroscedastic errors-in-variables model. AStA-Adv Stat Anal. 2014;98:165–195.
  • Yang SC, Li YM. Uniformly asymptotic normality of the regression weighted estimator for strong mixing samples. Acta Math Sin Chin Ser. 2006;49(5):1163–1170.
  • Hall P, Heyde CC. Martingale limit theory and its applications. New York (NY): Academic Press; 1980.
  • Petrov VV. Limit Theorems of Probability Theory. New York: Oxford University Press; 1995.

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