Almost sure and complete consistency of the estimator in nonparametric regression model for negatively orthant dependent random variables. (English) Zbl 1446.62116
Summary: In this paper, the author considers the nonparametric regression model with negatively orthant dependent random variables. The wavelet procedures are developed to estimate the regression function. For the wavelet estimator of unknown function \(g(\cdot)\), the almost sure consistency is derived and the complete consistency is established under the mild conditions. Our results generalize and improve some known ones for independent random variables and dependent random variables.
MSC:
| 62G08 | Nonparametric regression and quantile regression |
| 62G07 | Density estimation |
| 62G05 | Nonparametric estimation |
| 60E05 | Probability distributions: general theory |
| 60F15 | Strong limit theorems |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
Keywords:
negatively orthant dependent random variables; wavelet estimator; almost sure consistency; complete consistencyReferences:
| [1] | A. Antoniadis, G. Gregoire, and I. W. McKeague, Wavelet methods for curve estimation, J. Amer. Statist. Assoc. 89 (1994), no. 428, 1340-1353. · Zbl 0815.62018 · doi:10.1080/01621459.1994.10476873 |
| [2] | X. Bao, J. Lin, X. Wang, and Y. Wu, On complete convergence for weighted sums of arrays of rowwise END random variables and its statistical applications, Math. Slovaca 69 (2019), no. 1, 223-232. https://doi.org/10.1515/ms-2017-0216 · Zbl 1496.60026 · doi:10.1515/ms-2017-0216 |
| [3] | L. Ding, P. Chen, and Y. Li, Consistency for wavelet estimator in nonparametric regression model with extended negatively dependent samples, Statist. Papers (2018). https://doi.org/10.1007/s00362-018-1050-9 · Zbl 1461.62048 |
| [4] | L. Ding, Berry-Esseen bound of wavelet estimators in heteroscedastic regression model with random errors, Int. J. Comput. Math. 96 (2019), no. 4, 821-852. https://doi.org/10.1080/00207160.2018.1487958 · Zbl 1499.62129 · doi:10.1080/00207160.2018.1487958 |
| [5] | A. A. Georgiev, Local properties of function fitting estimates with application to system identification, in Mathematical statistics and applications, Vol. B (Bad Tatzmannsdorf, 1983), 141-151, Reidel, Dordrecht, 1985. · Zbl 0588.62059 |
| [6] | A. A. Georgiev, Consistent nonparametric multiple regression: the fixed design case, J. Multivariate Anal. 25 (1988), no. 1, 100-110. https://doi.org/10.1016/0047-259X(88)90155-8 · Zbl 0637.62044 · doi:10.1016/0047-259X(88)90155-8 |
| [7] | A. A. Georgiev and W. Greblicki, Nonparametric function recovering from noisy observations, J. Statist. Plann. Inference 13 (1986), no. 1, 1-14. https://doi.org/10.1016/0378-3758(86)90114-X · Zbl 0596.62041 · doi:10.1016/0378-3758(86)90114-X |
| [8] | P. Hall and P. Patil, Formulae for mean integrated squared error of nonlinear waveletbased density estimators, Ann. Statist. 23 (1995), no. 3, 905-928. https://doi.org/10.1214/aos/1176324628 · Zbl 0839.62042 · doi:10.1214/aos/1176324628 |
| [9] | K. Joag-Dev and F. Proschan, Negative association of random variables, with applications, Ann. Statist. 11 (1983), no. 1, 286-295. https://doi.org/10.1214/aos/1176346079 · Zbl 0508.62041 · doi:10.1214/aos/1176346079 |
| [10] | Y. Li, C. Wei, and G. Xing, Berry-Esseen bounds for wavelet estimator in a regression model with linear process errors, Statist. Probab. Lett. 81 (2011), no. 1, 103-110. https://doi.org/10.1016/j.spl.2010.09.024 · Zbl 1206.62073 · doi:10.1016/j.spl.2010.09.024 |
| [11] | X. Li, W. Z. Yang, S. H. Hu, and X. J. Wang, The Bahadur representation for sample quantile under NOD sequence, J. Nonparametr. Stat. 23 (2011), no. 1, 59-65. https://doi.org/10.1080/10485252.2010.486033 · Zbl 1359.62156 · doi:10.1080/10485252.2010.486033 |
| [12] | H.-Y. Liang, Asymptotic normality of wavelet estimator in heteroscedastic model with \({\alpha}\)-mixing errors, J. Syst. Sci. Complex. 24 (2011), no. 4, 725-737. https://doi.org/10.1007/s11424-010-8354-8 · Zbl 1255.93136 · doi:10.1007/s11424-010-8354-8 |
| [13] | H.-Y. Liang and B.-Y. Jing, Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences, J. Multivariate Anal. 95 (2005), no. 2, 227-245. https://doi.org/10.1016/j.jmva.2004.06.004 · Zbl 1070.62022 · doi:10.1016/j.jmva.2004.06.004 |
| [14] | H.-G. Muller, Weak and universal consistency of moving weighted averages, Period. Math. Hungar. 18 (1987), no. 3, 241-250. · Zbl 0596.62040 · doi:10.1007/BF01848087 |
| [15] | A. Shen, On the strong convergence rate for weighted sums of arrays of rowwise negatively orthant dependent random variables, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 107 (2013), no. 2, 257-271. https://doi.org/10.1007/s13398-012-0067-5 · Zbl 1278.60060 · doi:10.1007/s13398-012-0067-5 |
| [16] | A. Shen, Y. Zhang, and A. Volodin, Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables, Metrika 78 (2015), no. 3, 295-311. https://doi.org/10.1007/s00184-014-0503-y · Zbl 1333.60022 · doi:10.1007/s00184-014-0503-y |
| [17] | R. L. Taylor, R. F. Patterson, and A. Bozorgnia, A strong law of large numbers for arrays of rowwise negatively dependent random variables, Stochastic Anal. Appl. 20 (2002), no. 3, 643-656. https://doi.org/10.1081/SAP-120004118 · Zbl 1003.60032 · doi:10.1081/SAP-120004118 |
| [18] | X. Wang and S. Hu, On consistency of least square estimators in the simple linear EV model with negatively orthant dependent errors, Electron. J. Stat. 11 (2017), no. 1, 1434-1463. https://doi.org/10.1214/17-EJS1263 · Zbl 1362.62051 · doi:10.1214/17-EJS1263 |
| [19] | X. Wang, S. Hu, A. Shen, and W. Yang, An exponential inequality for a NOD sequence and a strong law of large numbers, Appl. Math. Lett. 24 (2011), no. 2, 219-223. https://doi.org/10.1016/j.aml.2010.09.007 · Zbl 1205.60068 · doi:10.1016/j.aml.2010.09.007 |
| [20] | X. Wang, S. Hu, and A. I. Volodin, Strong limit theorems for weighted sums of NOD sequence and exponential inequalities, Bull. Korean Math. Soc. 48 (2011), no. 5, 923-938. https://doi.org/10.4134/BKMS.2011.48.5.923 · Zbl 1234.60035 · doi:10.4134/BKMS.2011.48.5.923 |
| [21] | X. Wang, S. Hu, and W. Yang, Complete convergence for arrays of rowwise negatively orthant dependent random variables, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 106 (2012), no. 2, 235-245. https://doi.org/10.1007/s13398-011-0048-0 · Zbl 1260.60062 · doi:10.1007/s13398-011-0048-0 |
| [22] | X. Wang, S. Hu, W. Yang, and N. Ling, Exponential inequalities and inverse moment for NOD sequence, Statist. Probab. Lett. 80 (2010), no. 5-6, 452-461. https://doi.org/10.1016/j.spl.2009.11.023 · Zbl 1186.60015 · doi:10.1016/j.spl.2009.11.023 |
| [23] | X. Wang and Z. Si, Complete consistency of the estimator of nonparametric regression model under ND sequence, Statist. Papers 56 (2015), no. 3, 585-596. https://doi.org/10.1007/s00362-014-0598-2 · Zbl 1317.62029 · doi:10.1007/s00362-014-0598-2 |
| [24] | X. Wang, Y. Wu, S. Hu, and N. Ling, Complete moment convergence for negatively orthant dependent random variables and its applications in statistical models, Statist. Papers 2018 (2018). https://doi.prg/10.1007/s00362-018-0983-3 · Zbl 1447.60057 |
| [25] | Q. Y. Wu, Probability Limit Theory for Mixing and Dependent Sequences, Science Press of China, Beijing, 2006 |
| [26] | Q. Y. Wu, Complete convergence for weighted sums of sequences of negatively dependent random variables, J. Probab. Stat. 2011 (2011), Art. ID 202015, 16 pp. https://doi.org/10.1155/2011/202015 · Zbl 1221.60041 |
| [27] | L. G. Xue and Q. Liu, Bootstrap approximation of wavelet estimates in a semiparameter regression model, Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 4, 763-778. https://doi.org/10.1007/s10114-010-7236-2 · Zbl 1185.62080 · doi:10.1007/s10114-010-7236-2 |
| [28] | S. C. Yang, Maximal moment inequality for partial sums of strong mixing sequences and application, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 6, 1013-1024. https://doi.org/10.1007/s10114-005-0841-9 · Zbl 1121.60017 · doi:10.1007/s10114-005-0841-9 |
| [29] | R. Zhang, Y. Wu, W. F. Xu, and X. J. Wang, On complete consistency for the weighted estimator of nonparametric regression models, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113 (2019), no. 3, 2319-2333. https://doi.org/10.1007/s13398-018-00621-0 · Zbl 1426.62129 · doi:10.1007/s13398-018-00621-0 |
| [30] | X. · Zbl 1463.62296 · doi:10.1016/j.spl.2016.11.009 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
