Abstract
In this paper, the Berry–Esseen type bounds of the weighted estimator in a nonparametric regression model are investigated under some mild conditions when random errors are from a linear process generated by \(\varphi \)-mixing random variables. In particular, the rate of uniform normal approximation is near to \(O(n^{-\frac{3}{16}})\) by the choice of some constants, which generalizes and improves the corresponding results of Li et al. (Stat Probab Lett 81:103–110, 2011) and Ding et al. (J Inequal Appl 2018:10, 2018). Finally, the simulation study is provided to verify the validity of the theoretical results.
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Supported by the National Natural Science Foundation of China (11671012, 11871072, 11701004, 11701005), the Natural Science Foundation of Anhui Province (1808085QA03,1908085QA01), the Provincial Natural Science Research Project of Anhui Colleges (KJ2017A027, KJ2018B16), the Scientific Research Foundation Funded Project of Chuzhou University (2018qd01) and the Scientific Research Project of Chuzhou University (2015PY01, 2017qd16).
Appendix
Appendix
Lemma A.1
(cf. Liang and Fan 2009, Lemma 3.1) Let X and \(Y_1,Y_2,\ldots ,Y_m\) be random variables. Then for positive numbers \(a_1,a_2,\ldots ,a_m\), we have
Lemma A.2
(cf. Lu and Lin 1997, Lemma 1.2.8) Let \(\{X_n,n\ge 1\}\) be a sequence of \(\varphi \)-mixing random variables. Let \(X\in L_p({\mathcal {F}}_1^{k})\), \(Y\in L_q({\mathcal {F}}_{k+n}^{\infty })\), \(p\ge 1\), \(q\ge 1\) and \(1/p+1/q=1\). Then
Lemma A.3
(cf. Petrov 1995, Theorem 5.7) Let \(X_1,X_2,\ldots ,X_n\) be independent random variables with \(EX_j=0\) and \(E|X_j|^{2+\delta }<\infty \) for some \(0<\delta \le 1\) and \(j=1,2,\ldots ,n\). Denote \(B_n=\sum \nolimits _{j=1}^n \text {Var} X_j\), then
Lemma A.4
(cf. Yang 1995, Lemma 2) Let \(\{X_n,n\ge 1\}\) be a sequence of \(\varphi \)-mixing random variables with \(EX_j=0\). If \(\sum \nolimits _{j=1}^{\infty }\varphi ^{1/2}(j)<\infty \) and \(E|X_j|^q<\infty \) for \(q\ge 2\) and \(j=1,2,\ldots \), then for each \(n\ge 1\),
Lemma A.5
(cf. Li et al. 2008, Lemma 3.4) Let \(\{X_n,n\ge 1\}\) be a sequence of \(\varphi \)-mixing random variables. Suppose that p and q are two positive integers. Set \(\eta _l=:\sum \nolimits _{j=(l-1)(p+q)+1}^{(l-1)(p+q)+p}X_j\) for \(1\le l\le k\). Then
Proof of Lemma 5.1
It follows from the definition of \(S_{1n}^{''}\), Lemma A.4, \(Ee_0^2<\infty \) and (\(A_1\))–(\(A_4\)) that
Similar to (6.1), we have
and
This completes the proof of Lemma 5.1. \(\square \)
Proof of Lemma 5.2
It is easy to see that
thus,
Noting that \(ES_n^2=1\), we have
Hence, by \(C_r\) inequality, Hölder’s inequality, \(ES_n^2=1\) and Lemma 5.1, we have
From Lemma A.2, (\(A_1\)), (\(A_4\)) and (\(A_2\)), we have
Therefore, the desired result follows from (6.5) and (6.6) immediately. \(\square \)
Proof of Lemma 5.3
By Lemma A.3, we have
According to Lemma A.4, (\(A_1\)), (\(A_2\)) and (\(A_4\)), it follows that
Combined with (6.7) and Lemma 5.2, the proof of Lemma 5.3 is completed. \(\square \)
Proof of Lemma 5.4
Let \(\phi (t)\) and \(\psi (t)\) be the characteristic functions of \(S_{1n}^{'}\) and \(T_n\), respectively. By Esseen inequality (see Petrov 1995, Theorem 5.3), for any \(T>0\), we have
It can be found by Lemma A.5, (\(A_1\)), (\(A_2\)) and (\(A_4\)) that
Thus, \(L_{1n}\le C\gamma _{4n}T\). It is easily seen by Lemmas 5.3 and 5.2 that
Hence, \(L_{2n}\le C(\gamma _{2n}^{\delta /2}+\frac{1}{T}).\) By choosing \(T=\gamma _{4n}^{-1/2}\), we can obtain that
This completes the proof of the lemma. \(\square \)
Proof of Lemma 5.5
It follows by Lemma A.4, (\(A_1\))–(\(A_4\)) that
Similarly, we have
Thus, according to Markov’s inequality, (6.9), (6.10) and (6.3), we have
and
The proof is completed. \(\square \)
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Deng, X., Wang, X. & Wu, Y. The Berry–Esseen type bounds of the weighted estimator in a nonparametric model with linear process errors. Stat Papers 62, 963–984 (2021). https://doi.org/10.1007/s00362-019-01120-z
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DOI: https://doi.org/10.1007/s00362-019-01120-z