A Berry-Ess$\acute{e}$n bound of wavelet estimation for a nonparametric regression model under linear process errors based on LNQD sequence

Xueping Hu; Jingya Wang; Xueping Hu; Jingya Wang

doi:10.3934/math.2020448

AIMS Mathematics

2020, Volume 5, Issue 6: 6985-6995. doi: 10.3934/math.2020448

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Research article

A Berry-Ess$\acute{e}$n bound of wavelet estimation for a nonparametric regression model under linear process errors based on LNQD sequence

Xueping Hu ^,,
Jingya Wang

School of Mathematics and Physics, Anqing Normal University, Anhui, Anqing 246133, China

Received: 05 June 2020 Accepted: 02 September 2020 Published: 09 September 2020
MSC : 60G05, 62G20

By using some inequalities for linearly negative quadrant dependent random variables, Berry-Ess$\acute{e}$en bound of wavelet estimation for a nonparametric regression model is investigated under linear process errors based on linearly negative quadrant dependent sequence. The rate of uniform asymptotic normality is presented and the rate of convergence is near $O(n^{-\frac{1}{6}})$ under mild conditions, which generalizes or extends the corresponding results of Li et al.(2008) under associated random samples in some sense.
- wavelet estimation,
- Berry-Ess$\acute{e}$en bound,
- linearly negative quadrant dependent sequence
Citation: Xueping Hu, Jingya Wang. A Berry-Ess$\acute{e}$n bound of wavelet estimation for a nonparametric regression model under linear process errors based on LNQD sequence[J]. AIMS Mathematics, 2020, 5(6): 6985-6995. doi: 10.3934/math.2020448

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Abstract

By using some inequalities for linearly negative quadrant dependent random variables, Berry-Ess$\acute{e}$en bound of wavelet estimation for a nonparametric regression model is investigated under linear process errors based on linearly negative quadrant dependent sequence. The rate of uniform asymptotic normality is presented and the rate of convergence is near $O(n^{-\frac{1}{6}})$ under mild conditions, which generalizes or extends the corresponding results of Li et al.(2008) under associated random samples in some sense.

References

[1]	A. A. Georgiev, Local properties of function fitting estmates with application to system identification, Math. stat. Appl., 2 (1983), 141-151.
[2]	A. A. Georgiev, Consistent nonparametric multiple regression: The fixed design case, J. Multivariate Anal., 25 (1988), 100-110. doi: 10.1016/0047-259X(88)90155-8
[3]	G. G. Roussas, L. T. Tran, D. A. Ioannides, Fixed design regression for time series: Asymptotic normality, J. Multivariate Anal., 40 (1992), 262-291. doi: 10.1016/0047-259X(92)90026-C
[4]	S. Yang, Uniformly asymptotic normality of the regression weighted estimator for negatively associated samples, Stat. Probabil. Lett., 62 (2003), 101-110. doi: 10.1016/S0167-7152(02)00427-3
[5]	L. Tran, G. Roussas, S. Yakowitz, et al. Fixed-design regression for linear time series, Ann. Stat., 24 (1996), 975-991.
[6]	H. Y. Liang, Y. Y. Li, A Berry-Esseen type bound of regression estimator based on linear process errors, J. Korean Math. Soc., 45 (2008), 1753-1767. doi: 10.4134/JKMS.2008.45.6.1753
[7]	A. Antoniadis, G. Gregoire, I. W. Mckeague, Wavelet methods for curve estimation, J. Am. Stat. Assoc., 89 (1994), 1340-1352. doi: 10.1080/01621459.1994.10476873
[8]	H. Y. Liang, Y. Y. Qi, Asymptotic normality of wavelet estimator of regression function under NA assumption, B. Korean Math. Soc., 44 (2007), 247-257. doi: 10.4134/BKMS.2007.44.2.247
[9]	L. G. Xue, Uniform convergence rates of the wavelet estimator of regression function under mixing error, Acta Math. Sci., 22 (2002), 528-535.
[10]	Y. Sun, G. X. Chai, Nonparametric wavelet estimation of a fiexed designed regression function, Acta Math. Sci., 24 (2004), 597-606. doi: 10.1016/S0252-9602(17)30242-4
[11]	Y. Li, S. Yang, Y. Zhou, Consistency and uniformly asymptotic normality of wavelet estimator in regression model with associated samples, Stat. Probabil. Lett., 78 (2008), 2947-2956. doi: 10.1016/j.spl.2008.05.004
[12]	Y. Li, C. Wei, G. Xing, Berry-Esseen bounds for wavelet estimator in a regression model with linear process errors, Stat. Probabil. Lett., 81 (2011), 103-110. doi: 10.1016/j.spl.2010.09.024
[13]	X. Zhou, J. Lin, C. M. Yin, Asymptotic properties of wavelet-based estimator in nonparametric regression model with weakly dependent process, J. Inequal. Appl., 2013 (2013), 1-18. doi: 10.1186/1029-242X-2013-1
[14]	K. Joag-Dev, F. Proschan, Negative association of random variables with applications, Ann. Stat., 11 (1983), 286-295. doi: 10.1214/aos/1176346079
[15]	E. L. Lehmann, Some concepts of dependence, Ann. Math. Stat., 37 (1996), 1137-1153.
[16]	C. M. Newman, Asymptotic independent and limit theorems for positively and negatively dependent random variables, Inequalities in statistics and probability, 5 (1984), 127-140.
[17]	X. Wang, S. Hu, W. Yang, et al. Exponential inequalities and complete convergence for a LNQD sequence, J. Korean Stat. Soc., 39 (2010), 555-564. doi: 10.1016/j.jkss.2010.01.002
[18]	Y. Li, J. Guo, N. Li, Some inequalities for a LNQD sequence with applications, J. Inequal. Appl., 2012 (2012), 1-9. doi: 10.1186/1029-242X-2012-1
[19]	L. Ding, P. Chen, Y. Li, Berry-Esseen bounds of weighted kernel estimator for a nonparametric regression model based on linear process errors under a LNQD sequence, J. Inequal. Appl., 2018 (2018), 1-12. doi: 10.1186/s13660-017-1594-6
[20]	V. V. Petrov, Limit Theorem of Probability Theory: Sequences of Independent Random Variables, Oxford Science Publications, Clarendon Press, Oxford, 1995.
[21]	L. Zhang, A functional central limit theorem for asymptotically negatively dependent random fields, Acta. Math. Hung., 86 (2000), 237-259. doi: 10.1023/A:1006720512467

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