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Complete consistency of estimators for regression models based on extended negatively dependent errors

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Abstract

In this paper, we investigate the consistency of the estimators of nonparametric regression model and multiple linear regression model based on extended negatively dependent errors. The complete convergence rates of the estimators of nonparametric regression model are presented. In addition, therth-mean consistency and complete consistency of the least squares estimator of the multiple linear regression model are obtained too. Finally, some examples and some simulations are illustrated.

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References

  • Adler A, Rosalsky A (1987) Some general strong laws for weighted sums of stochastically dominated random variables. Stoch Anal Appl 5(1):1–16

    Article MathSciNet MATH  Google Scholar 

  • Adler A, Rosalsky A, Taylor RL (1989) Strong laws of large numbers for weighted sums of random elements in normed linear spaces. Int J Math Math Sci 12(3):507–530

    Article MathSciNet MATH  Google Scholar 

  • Anderson TW (2003) An introduction to multivariate statistical analysis, 3rd edn. John Wiley & Sons, New York

    MATH  Google Scholar 

  • Baltagi BH, Krämer W (1994) Consistency, asymptotic unbiasedness and bounds on the bias of\(s^2\) in the linear regression model with error component disturbances. Stat Papers 35(1):323–328

    Article MathSciNet MATH  Google Scholar 

  • Bulinski AV, Shaskin A (2007) Limit theorems for associated random fields and related systems. World Scientific, Singapore

    Book  Google Scholar 

  • Chen GJ, Lai TL, Wei CZ (1981) Convergence systems and strong consistency of least squares estimates in regression models. J Multivariate Anal 11(3):319–333

    Article MathSciNet MATH  Google Scholar 

  • Chen XR (1981) Introduction to mathematical statistics. Science Press, Beijing (In Chinese)

    Google Scholar 

  • Chen XR, Chen GJ, Wu QG, Zhao LC (2010) Theory of parametric estimation for linear model. Science Press, Beijing (In Chinese)

    Google Scholar 

  • Chen ZY, Wang HB, Wang XJ (2016) The consistency for the estimator of nonparametric regression model based on martingale difference errors. Stat Papers 57(2):451–469

  • Erdös P (1949) On a Theorem of Hsu and Robbins. Ann Math Stat 20(2):286–291

    Article MathSciNet MATH  Google Scholar 

  • Georgiev AA, Greblicki W (1986) Nonparametric function recovering from noisy observations. J Stat Plan Inference 13(1):1–14

    Article MathSciNet MATH  Google Scholar 

  • Georgiev AA (1988) Consistent nonparametric multiple regression: the fixed design case. J Multivariate Anal 25(1):100–110

    Article MathSciNet MATH  Google Scholar 

  • Hanson DL, Wright FT (1971) A bound on tail probabilities for quadratic forms in independent random variables. Ann Math Stat 42(3):1079–1083

    Article MathSciNet MATH  Google Scholar 

  • Hu SH, Zhu CH, Chen YB (2002) Fixed-design regression for linear time series. Acta Math Sci Ser B Engl Ed 22(1):9–18

    Article MathSciNet MATH  Google Scholar 

  • Hu SH, Pan GM, Gao QB (2003) Estimate problem of regression models with linear process errors. Appl Math J Chin Univ 18A(1):81–90

    MATH  Google Scholar 

  • Hu TC, Chiang CY, Taylor RL (2009) On complete convergence for arrays of rowwise m-negatively associated random variables. Nonlinear Anal 71(12):e1075–e1081

    Article MathSciNet MATH  Google Scholar 

  • Hu TC, Rosalsky A, Wang KL (2015) Complete convergence theorems for extended negatively dependent random variables. Sankhya 77A(1):1–29

    MathSciNet MATH  Google Scholar 

  • Hsu PL, Robbins H (1947) Complete convergence and the law of large numbers. Proc Natl Acad Sci USA 33(2):25–31

    Article MathSciNet MATH  Google Scholar 

  • Joag-Dev K, Proschan F (1983) Negative association of random variables with applications. Ann Stat 11(1):286–295

    Article MathSciNet MATH  Google Scholar 

  • Ko B, Tang Q (2008) Sums of dependent nonnegative random variables with subexponential tails. J Appl Probab 45(1):85–94

    Article MathSciNet MATH  Google Scholar 

  • Lehmann EL (1966) Some concepts of dependence. Ann Math Stat. 37(5):1137–1153

    Article MathSciNet MATH  Google Scholar 

  • Liang HY, Jing BY (2005) Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences. J Multivariate Anal 95(2):227–245

    Article MathSciNet MATH  Google Scholar 

  • Liu L (2009) Precise large deviations for dependent random variables with heavy tails. Stat Probab Lett 79(9):1290–1298

    Article MathSciNet MATH  Google Scholar 

  • Liu L (2010) Necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails. Sci China Ser A 53(6):1421–434

    Article MathSciNet MATH  Google Scholar 

  • Miao Y, Zhao FF, Wang K, Chen YP (2013) Asymptotic normality and strong consistency of LS estimators in the EV regression model with NA errors. Stat Pap 54(1):193–206

    Article MathSciNet MATH  Google Scholar 

  • Müller HG (1987) Weak and universal consistency of moving weighted averages. Period Math Hung 18(3):241–250

    Article MathSciNet MATH  Google Scholar 

  • Oliveira PE (2012) Asymptotics for associated random variables. Springer, Berlin

    Book MATH  Google Scholar 

  • Prakasa Rao BLS (2012) Associated sequences, demimartingales and nonparametric inference. Birkhäuser, Springer, Basel

    Book MATH  Google Scholar 

  • Roussas GG (1989) Consistent regression estimation with fixed design points under dependence conditions. Stat Probab Lett 8(1):41–50

    Article MathSciNet MATH  Google Scholar 

  • Roussas GG, Tran LT, Ioannides DA (1992) Fixed design regression for time series: asymptotic normality. J Multivariate Anal 40(2):262–291

    Article MathSciNet MATH  Google Scholar 

  • Shen AT (2011) Probability inequalities for END sequence and their applications. J Inequal Appl 1:1–12

    MathSciNet  Google Scholar 

  • Shen AT (2013). Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models. Abstr Appl Anal 2013, Article ID 862602, 9 pages

  • Song SH (1994) Consistency and asymptotic unbiasedness of S2 in the serially correlated error components regression model for panel data. Stat Pap 37(3):267–275

    Article  Google Scholar 

  • Sung SH (2011) On the exponential inequalities for negatively dependent random variables. J Math Anal Appl 381(2):538–545

    Article MathSciNet MATH  Google Scholar 

  • Tran L, Roussas G, Yakowitz S, Van BT (1996) Fixed-design regression for linear time series. Ann Stat 24(3):975–991

    Article MathSciNet MATH  Google Scholar 

  • Wang XJ, Ge MM, Hu SH, Wang XZ (2013a) The strong consistency of the estimator of fixed-design regression model under negatively dependent sequences. Abstr Appl Anal 2013, Article ID 521618, 7 pages

  • Wang XJ, Hu SH, Yang WZ, Ling NX (2010) Exponential inequalities and inverse moment for NOD sequence. Stat Probab Lett 80(5–6):452–461

    Article MathSciNet MATH  Google Scholar 

  • Wang XJ, Hu TC, Volodin AI, Hu SH (2013b) Complete convergence for weighted sums and arrays of rowwise extended negatively dependent random variables. Commun Stat Theory Methods 42(13):2391–2401

    Article MathSciNet MATH  Google Scholar 

  • Wang XJ, Si ZY (2015) Complete consistency of the estimator of nonparametric regression model under ND sequence. Stat Pap 56(3):585–596

    Article MathSciNet MATH  Google Scholar 

  • Wang XJ, Wang SJ, Hu SH, Ling JM, Wei YF (2013c) On complete convergence for weighted sums of rowwise extended negatively dependent random variables. Stoch Int J Probab Stoch Process 85(6):1060–1072

    Article MathSciNet MATH  Google Scholar 

  • Wang XJ, Xia FX, Ge MM, Hu SH, Yang WZ (2012) Complete consistency of the estimator of nonparametric regression models based on\(\tilde{\rho }\)-mixing sequences. Abstr Appl Anal 2012 Article ID 907286, 12 pages

  • Wang XJ, Xu C, Hu TC, Volodin A, Hu SH (2014) On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST 23(3):607–629

    Article MathSciNet MATH  Google Scholar 

  • Wang XJ, Zheng LL, Xu C, Hu SH (2015) Complete consistency for the estimator of nonparametric regression models based on extended negatively dependent errors. Statistics 49(2):396–407

    Article MathSciNet MATH  Google Scholar 

  • Wright FT (1973) A bound on tail probabilities for quadratic forms in independent random variables whose distributions are not necessarily symmetric. Ann Probab 1(6):1068–1070

    Article MathSciNet MATH  Google Scholar 

  • Wu QY (2010) Complete convergence for negatively dependent sequences of random variables. J Inequal Appl 2010, Article ID 507293, 10 pages

  • Wu QY (2011) Complete convergence for weighted sums of sequences of negatively dependent random variables. J Probab Stat 2011, Article ID 202015, 16 pages

  • Wu QY (2012) A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables. J Inequal Appl 2012:50

    Article MathSciNet MATH  Google Scholar 

  • Wu QY, Chen PY (2014) A Berry-Esseen Type Bound in Kernel Density Estimation for Negatively Associated Censored Data. J Appl Math 2013 Article ID 541250, 9 pages

  • Yang WZ, Wang XJ, Wang XH, Hu SH (2012) The consistency for estimator of nonparametric regression model based on NOD errors. J Inequal Appl 2012, 2012:140, 13 pages

Download references

Acknowledgments

The authors are deeply grateful to Editor and anonymous referees for their careful reading and insightful comments, which helped in improving the earlier version of this paper. This work is supported by the National Natural Science Foundation of China (11426032, 11501004, 11501005), Natural Science Foundation of Anhui Province (1408085QA02, 1508085J06, 1608085QA02), Science Research Project of Anhui Colleges (KJ2014A020), Quality Engineering Project of Anhui Province (2015jyxm054) and Applied Teaching Model Curriculum of Anhui University (XJYYKC1401, ZLTS2015053).

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Authors and Affiliations

  1. School of Mathematical Science, Anhui University, Hefei, 230601, People’s Republic Of China

    Wenzhi Yang, Haiyun Xu, Ling Chen & Shuhe Hu

Authors
  1. Wenzhi Yang

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  2. Haiyun Xu

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  3. Ling Chen

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  4. Shuhe Hu

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Corresponding author

Correspondence toShuhe Hu.

Appendix

Appendix

Lemma A

(cf. Wang et al. (2015), Lemma 2.2). Let\(\{Z_n,n\ge 1\}\) be a sequence of END random variables such that\(EZ_n=0\) and\(|Z_n|\le d_n\)a.s. for each\(n\ge 1\), where\(\{d_n,n\ge 1\}\) is a sequence of positive constants. Assume that\(t>0\) such that\(t\max \limits _{1\le i\le n}d_i\le 1\). Then for every\(\varepsilon >0\), there exists a constant\(M>0\) such that

$$\begin{aligned} P\left( \left| \sum \limits _{i=1}^n Z_i\right| \ge \varepsilon \right) \le 2M \exp \left\{ -t\varepsilon +t^2\sum \limits _{i=1}^nEZ_i^2\right\} , \quad n\ge 1. \end{aligned}$$
(11)

Corollary A

Let\(\{Z_n,n\ge 1\}\) be a sequence of END random variables such that\(EZ_n=0\) and\(|Z_n|\le d_n\)a.s. for each\(n\ge 1\), where\(\{d_n,n\ge 1\}\) is a sequence of positive constants. Denote\(b_n=\max \limits _{1\le i\le n}d_i\) and\(\Delta _n^2=\sum _{i=1}^n EZ_i^2\) for each\(n\ge 1\). Then for every\(\varepsilon >0\), there exists a constant\(M>0\) such that

$$\begin{aligned} P\left( \left| \sum \limits _{i=1}^n Z_i\right| \ge \varepsilon \right) \le 2M\exp \left\{ -\frac{\varepsilon ^2}{2(2\Delta _n^2+b_n\varepsilon )}\right\} , \quad n\ge 1. \end{aligned}$$
(12)

Proof

Taking\(t=\frac{\varepsilon }{2\Delta _n^2+b_n\varepsilon }\) in (11) of Lemma A, we immediately obtain (12).\(\square \)

Lemma B

(cf. Adler and Rosalsky (1987), Lemma 1, and Adler et al. (1989), Lemma 3). Let\(\{Z_n,n\ge 1\}\) be a sequence of random variables, which is stochastically dominated by a nonnegative random variableZ. Then, for any\(\alpha >0\) and\(b>0\), the following two statements hold:

$$\begin{aligned} E\big [|Z_n|^{\alpha }I(|Z_n|\le b)\big ]\le C_1\left\{ E[Z^{\alpha }I(X\le b)]+b^{\alpha }P(Z>b)\right\} , \end{aligned}$$
$$\begin{aligned} E\big [|Z_n|^{\alpha }I(|Z_n|>b)\big ]\le C_2E\big [Z^{\alpha }I(Z>b)\big ], \end{aligned}$$

where\(C_1\) and\(C_2\) are positive constants. Consequently, it has\(E|Z_n|^{\alpha }\le C_3EZ^{\alpha }\) for all\(n\ge 1\).

Proof of Theorem 2.1:

In view of the proof of (4.5) in Yang et al. (2012), by the local Lipschitz condition ofg(x) and the assumptions\((H_1)\)–\((H_3)\), one can obtain that

$$\begin{aligned} |E\hat{g}_n(x)-g(x)|=O\left( n^{-\frac{1}{2r}}\log n\right) , \quad x\in A, \quad \text {as} \quad n\rightarrow \infty . \end{aligned}$$
(13)

With\(x\in A\), to prove (6), we have to prove that

$$\begin{aligned} |\hat{g}_n(x)-E\hat{g}_n(x)|=\left| \sum \limits _{i=1}^n W_{ni}(x)\varepsilon _{ni}\right| =O\left( n^{-\frac{1}{2r}}\log n\right) , \quad \text {completely},\quad \text {as}\quad n\rightarrow \infty . \end{aligned}$$
(14)

In view of Lemma 3.1 of Liu (2010), for the fixedx, we can see that\(\{W_{ni}^{+}(x)\varepsilon _i,1\le i\le n\}\) and\(\{W_{ni}^{-}(x)\varepsilon _i,1\le i\le n\}\) are also sequences of END random variables. Noting that\(W_{ni}(x)\varepsilon _i=W_{ni}^{+}(x)\varepsilon _i-W_{ni}^{-}(x)\varepsilon _i\), without loss of generality, we assume\(W_{ni}(x)\ge 0\) in the proof. For all\(n\ge 1\) and\(1\le i\le n\), let

$$\begin{aligned} \varepsilon ^{(n)}_{1,i}= & {} -n^{\frac{1}{2r}}I\left( \varepsilon _{ni}<-n^{\frac{1}{2r}}\right) +\varepsilon _{ni}I\left( |\varepsilon _{ni}|\le n^{\frac{1}{2r}}\right) +n^{\frac{1}{2r}}I\left( \varepsilon _{ni}>n^{\frac{1}{2r}}\right) ,\\ \varepsilon ^{(n)}_{2,i}= & {} \varepsilon _{ni}-\varepsilon ^{(n)}_{1,i}=\left( \varepsilon _{ni}-n^{\frac{1}{2r}}\right) I\left( \varepsilon _{ni}>n^{\frac{1}{2r}}\right) +\left( \varepsilon _{ni}+n^{\frac{1}{2r}}\right) I\left( \varepsilon _{ni}<-n^{\frac{1}{2r}}\right) ,\\ \varepsilon _{1,i}(n)= & {} -n^{\frac{1}{2r}}I\left( \varepsilon _{i}<-n^{\frac{1}{2r}}\right) +\varepsilon _{i}I\left( |\varepsilon _{i}|\le n^{\frac{1}{2r}}\right) +n^{\frac{1}{2r}}I\left( \varepsilon _{i}>n^{\frac{1}{2r}}\right) ,\\ \varepsilon _{2,i}(n)= & {} \varepsilon _{i}-\varepsilon _{1,i}(n)=\left( \varepsilon _{i}-n^{\frac{1}{2r}}\right) I\left( \varepsilon _{i}>n^{\frac{1}{2r}}\right) +\left( \varepsilon _{i}+n^{\frac{1}{2r}}\right) I\left( \varepsilon _{i}<-n^{\frac{1}{2r}}\right) . \end{aligned}$$

Since\(E\varepsilon _{ni}=E\varepsilon _i=0\) for\(1\le i\le n\) and\(n\ge 1\), it can be argued that

$$\begin{aligned} \hat{g}_n(x)-E\hat{g}_n(x)= & {} \sum \limits _{i=1}^n W_{ni}(x)\varepsilon _{ni}\nonumber \\= & {} \sum \limits _{i=1}^nW_{ni}(x)\left[ \varepsilon ^{(n)}_{1,i}-E\varepsilon ^{(n)}_{1,i}\right] +\sum \limits _{i=1}^n W_{ni}(x)\left[ \varepsilon ^{(n)}_{2,i}-E\varepsilon ^{(n)}_{2,i}\right] \nonumber \\:= & {} T_{n1}+T_{n2}. \end{aligned}$$
(15)

Obviously,\(\{W_{ni}(x)(\varepsilon _{1,i}(n)-E\varepsilon _{1,i}(n))\}_{1\le i\le n}\) are END random variables with mean zero. From Lemma B,\((H_2)\) and\(EZ^{2r+2}<\infty \) (\(r\ge 1\)), it follows

$$\begin{aligned}&\max \limits _{1\le i\le n}\left| W_{ni}(x)(\varepsilon _{1,i}(n)-E\varepsilon _{1,i}(n))\right| \le 2n^{\frac{1}{2r}}\max \limits _{1\le i\le n}|W_{ni}(x)|\le c_1n^{-\frac{1}{2r}},\\&\quad \sum \limits _{i=1}^n E\left[ W_{ni}(x)(\varepsilon _{1,i}(n)-E\varepsilon _{1,i}(n))\right] ^2\le \sum \limits _{i=1}^nW^2_{ni}(x) E\varepsilon ^2_{1,i}(n)\\&\qquad \le \sum \limits _{i=1}^nW^2_{ni}(x) E\varepsilon _{i}^2\le c_2EZ^2\max \limits _{1\le i\le n}|W_{ni}(x)|\sum \limits _{i=1}^n|W_{ni}(x)| \le c_3n^{-\frac{1}{r}}. \end{aligned}$$

Since that\((\varepsilon _{n1},\varepsilon _{n2},\ldots ,\varepsilon _{nn})\) has the same distribution as\((\varepsilon _{1},\varepsilon _{2},\ldots ,\varepsilon _{n})\) for eachn, we apply Corollary A and obtain that for a sufficiently large\(C>0\),

$$\begin{aligned}&\sum \limits _{n=1}^\infty P\left( |T_{n1}|\ge C n^{-\frac{1}{2r}}\log n\right) \nonumber \\&\quad =\sum \limits _{n=1}^\infty P\left( \left| \sum \limits _{i=1}^nW_{ni}(x)\left[ \varepsilon ^{(n)}_{1,i}-E\varepsilon ^{(n)}_{1,i}\right] \right| \ge Cn^{-\frac{1}{2r}}\log n\right) \nonumber \\&\quad =\sum \limits _{n=1}^\infty P\left( \left| \sum \limits _{i=1}^nW_{ni}(x)\left[ \varepsilon _{1,i}(n)-E\varepsilon _{1,i}(n)\right] \right| \ge C n^{-\frac{1}{2r}}\log n\right) \nonumber \\&\quad \le c_4\sum \limits _{n=1}^\infty \exp \left\{ -\frac{ C^2 n^{-\frac{1}{r}}\log ^2 n}{2\left( 2c_3n^{-\frac{1}{r}}+c_1 C n^{-\frac{1}{r}}\log n\right) }\right\} \nonumber \\&\quad \le c_5\sum \limits _{n=1}^\infty \frac{1}{n^{\frac{C}{3c_1}}}<\infty . \end{aligned}$$
(16)

Meanwhile, by\((H_2)\), Lemma B and\(EZ^{2r+2}<\infty \), it yields

$$\begin{aligned}&\sum \limits _{n=2}^\infty P\left( |T_{n2}|\ge n^{-\frac{1}{2r}}\log n\right) \nonumber \\&\quad =\sum \limits _{n=2}^\infty P\left( \left| \sum \limits _{i=1}^n W_{ni}(x)\left[ \varepsilon ^{(n)}_{2,i}-E\varepsilon ^{(n)}_{2,i}\right] \right| \ge n^{-\frac{1}{2r}}\log n\right) \nonumber \\&\quad =\sum \limits _{n=2}^\infty P\left( \left| \sum \limits _{i=1}^n W_{ni}(x)\left[ \varepsilon _{2,i}(n)-E\varepsilon _{2,i}(n)\right] \right| \ge n^{-\frac{1}{2r}}\log n\right) \nonumber \\&\quad \le \sum \limits _{n=2}^\infty \frac{c_1}{\log n}n^{\frac{1}{2r}}\sum \limits _{i=1}^n |W_{ni}(x)|E\left( |\varepsilon _{i}|I\left( |\varepsilon _i|>n^{\frac{1}{2r}}\right) \right) \nonumber \\&\quad \le \sum \limits _{n=1}^\infty c_2n^{\frac{1}{2r}} E\left( ZI\left( Z>n^{\frac{1}{2r}}\right) \right) \nonumber \\&\quad =c_2\sum \limits _{n=1}^\infty n^{\frac{1}{2r}}\sum \limits _{k=n}^\infty E\left( ZI\left( k^{\frac{1}{2r}}<Z\le (k+1)^{\frac{1}{2r}}\right) \right) \nonumber \\&\quad =c_2\sum \limits _{k=1}^\infty E\left( ZI\left( k^{\frac{1}{2r}}<Z\le (k+1)^{\frac{1}{2r}}\right) \right) \sum \limits _{n=1}^kn^{\frac{1}{2r}}\nonumber \\&\quad \le c_3\sum \limits _{k=1}^\infty k^{\frac{1}{2r}+1} E\left( ZI\left( k^{\frac{1}{2r}}<Z\le (k+1)^{\frac{1}{2r}}\right) \right) \nonumber \\&\quad \le c_3\sum \limits _{k=1}^\infty E\left( Z^{2r+2}I\left( k^{\frac{1}{2r}}<Z\le (k+1)^{\frac{1}{2r}}\right) \right) \nonumber \\&\quad \le c_3EZ^{2r+2}<\infty . \end{aligned}$$
(17)

Consequently, by (15), (16), (17), (14) is completely proved. The desired result (6) follows from (13) and (14) immediately.\(\square \)

Proof of Corollary 2.1

It suffices to show that the conditions of Theorem2.1 are satisfied. For every\(x\in [0,1]\), we can argue by definition of\(R_i(x)\) and choice of\(x_{ni}\),\(k_n=\lceil n^{1/r}\rceil \) and (7) that

$$\begin{aligned} \sum \limits _{i=1}^n \tilde{W}_{ni}(x)=\sum \limits _{i=1}^n W_{nR_i(x)}(x)=\sum \limits _{i=1}^{k_n} \frac{1}{k_n}=1, \end{aligned}$$
$$\begin{aligned} \max \limits _{1\le i\le n}\tilde{W}_{ni}(x)=\frac{1}{k_n}\le c_1n^{-\frac{1}{r}}. \end{aligned}$$

For\(r>3/2\), we have by taking\(l>1/(2r-3)\) that

$$\begin{aligned}&\sum \limits _{i=1}^n \tilde{W}_{ni}(x)I\left( |x_{ni}-x|>n^{-\frac{1}{2r}}\log n\right) \le \sum \limits _{i=1}^n \tilde{W}_{ni}(x)\frac{|x_{ni}-x|^l}{|n^{-\frac{1}{2r}}\log n|^l}\\&\quad =\sum \limits _{i=1}^{k_n} \frac{1}{k_n}\frac{\left| x^{(n)}_{R_i(x)}-x\right| ^l}{\log ^{l}n}n^{\frac{l}{2r}}\le \sum \limits _{i=1}^{k_n} \frac{1}{k_n}\frac{\big (\frac{i}{n}\big )^l}{\log ^{l}n}n^{\frac{l}{2r}}\\&\quad \le \frac{\big (\frac{k_n}{n}\big )^l}{\log ^{l}n}n^{\frac{l}{2r}}\le \frac{c_1}{n^{l(1-\frac{3}{2r})}\log ^{l}n} \le \frac{c_2}{n^{\frac{1}{2r}}\log ^{l} n}. \end{aligned}$$

Since thatg(x) satisfies the Lipschitz condition for\(x\in [0,1]\), it can be seen that the assumptions of\((H_1)\)–\((H_3)\) are satisfied. Consequently, Corollary2.1 follows from Theorem2.1 immediately.\(\square \)

Proof of Theorem 2.2:

Obviously, it has\((b_{ni}^{(j)})\varepsilon _i=\big (b_{ni}^{(j)}\big )^{+}\varepsilon _i-\big (b_{ni}^{(j)}\big )^{-}\varepsilon _i\). Furthermore,\(\left\{ (b_{ni}^{(j)})^{+}\varepsilon _i,1\le i\le n\right\} \) and\(\left\{ (b_{ni}^{(j)})^{-}\varepsilon _i,1\le i\le n\right\} \) are also END random variables. Without loss of generality, we assume\(b_{ni}^{(j)}\ge 0\) in this proof too. On the one hand, for\(r\ge 2\) and\(j\in \{1,2,\ldots p\}\), by\(EZ^r<\infty \), (5),\(b_n^{(j)}\rightarrow \infty \), Corollary 3.2 of Shen (2011) and Lemma B, we obtain that

$$\begin{aligned} E|\hat{\beta }_{nj}-\beta _{j}|^r= & {} E\left| \sum \limits _{i=1}^n\frac{b_{ni}^{(j)}\varepsilon _i}{b_n^{(j)}}\right| ^r =\frac{1}{(b_{n}^{(j)})^{r}}E\left| \sum \limits _{i=1}^nb_{ni}^{(j)}\varepsilon _i\right| ^r\nonumber \\\le & {} \frac{C_1}{(b_{n}^{(j)})^{r}}\left\{ \sum \limits _{i=1}^n|b_{ni}^{(j)}|^rE|\varepsilon _i|^r +\left( \sum \limits _{i=1}^n(b_{ni}^{(j)})^2E\varepsilon _i^2\right) ^{r/2}\right\} \nonumber \\\le & {} \frac{C_2EZ^r}{(b_{n}^{(j)})^{r}}\left\{ \sum \limits _{i=1}^n|b_{ni}^{(j)}|^r +\left( \sum \limits _{i=1}^n(b_{ni}^{(j)})^2\right) ^{r/2}\right\} \nonumber \\\le & {} \frac{C_3EZ^r}{(b_{n}^{(j)})^{r}}\left( \sum \limits _{i=1}^n(b_{ni}^{(j)})^2\right) ^{r/2} =\frac{C_3EZ^r}{(b_{n}^{(j)})^{r/2}}\rightarrow 0 \quad as \quad n\rightarrow \infty ,\qquad \end{aligned}$$
(18)

where the third inequality uses the fact that\(\left( \sum _{i=1}^n a_i^\alpha \right) ^{1/\alpha }\ge \left( \sum _{i=1}^n a_i^\beta \right) ^{1/\beta }\) for any positive number sequence\(\{a_i,1\le i\le n\}\) and\(1\le \alpha \le \beta \). Therefore, (9) follows from (18).

On the other hand, by Markov’s inequality,\(\sum \nolimits _{n=1}^{\infty }(b_n^{(j)})^{-r/2}<\infty \) and (18), one obtains that for any\(\lambda >0\),

$$\begin{aligned} \sum \limits _{n=1}^\infty P(|\hat{\beta }_{nj}-\beta _{j}|>\lambda )\le \sum \limits _{n=1}^\infty \frac{E|\hat{\beta }_{nj}-\beta _{j}|^r}{\lambda ^r}\le \frac{C_1}{\lambda ^r}\sum \limits _{n=1}^\infty \frac{1}{(b_{n}^{(j)})^{r/2}}<\infty . \end{aligned}$$

Thus, (10) is completely proved.\(\square \)

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Yang, W., Xu, H., Chen, L.et al. Complete consistency of estimators for regression models based on extended negatively dependent errors.Stat Papers59, 449–465 (2018). https://doi.org/10.1007/s00362-016-0771-x

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