×

The strong consistency of the estimator of fixed-design regression model under negatively dependent sequences. (English) Zbl 1364.62100

Summary: We study the strong consistency of estimator of fixed design regression model under negatively dependent sequences by using the classical Rosenthal-type inequality and the truncated method. As an application, the strong consistency for the nearest neighbor estimator is obtained.

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Georgiev, A. A.; Grossmann, W., Local properties of function fitting estimates with applications to system identification, Mathematical Statistics and Applications. Mathematical Statistics and Applications, Volume 2 of Proceedings of the 4th Pannonian Symposium on Mathematical Statistics, Bad Tatzmannsdorf, Austria, 4-10 September 1983, 141-151 (1985), Dordrecht, The Netherlands: Reidel, Dordrecht, The Netherlands
[2] Georgiev, A. A.; Greblicki, W., Nonparametric function recovering from noisy observations, Journal of Statistical Planning and Inference, 13, 1, 1-14 (1986) · Zbl 0596.62041
[3] Georgiev, A. A., Consistent nonparametric multiple regression: the fixed design case, Journal of Multivariate Analysis, 25, 1, 100-110 (1988) · Zbl 0637.62044
[4] Müller, H. G., Weak and universal consistency of moving weighted averages, Periodica Mathematica Hungarica, 18, 3, 241-250 (1987) · Zbl 0596.62040
[5] Fan, Y., Consistent nonparametric multiple regression for dependent heterogeneous processes: the fixed design case, Journal of Multivariate Analysis, 33, 1, 72-88 (1990) · Zbl 0698.62040
[6] Roussas, G. G., Consistent regression estimation with fixed design points under dependence conditions, Statistics & Probability Letters, 8, 1, 41-50 (1989) · Zbl 0674.62026
[7] Roussas, G. G.; Tran, L. T.; Ioannides, D. A., Fixed design regression for time series: asymptotic normality, Journal of Multivariate Analysis, 40, 2, 262-291 (1992) · Zbl 0764.62073
[8] Tran, L.; Roussas, G.; Yakowitz, S.; Truong Van, B., Fixed-design regression for linear time series, The Annals of Statistics, 24, 3, 975-991 (1996) · Zbl 0862.62069
[9] Hu, S. H.; Zhu, C. H.; Chen, Y. B.; Wang, L. C., Fixed-design regression for linear time series, Acta Mathematica Scientia B, 22, 1, 9-18 (2002) · Zbl 1010.62085
[10] Hu, S. H.; Pan, G. M.; Gao, Q. B., Estimation problems for a regression model with linear process errors, Applied Mathematics-A Journal of Chinese Universities, 18, 1, 81-90 (2003)
[11] Liang, H.-Y.; Jing, B.-Y., Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences, Journal of Multivariate Analysis, 95, 2, 227-245 (2005) · Zbl 1070.62022
[12] Yang, W. Z.; Wang, X. J.; Wang, X. H.; Hu, S. H., The consistency for estimator of nonparametric regression model based on NOD errors, Journal of Inequalities and Applications, 2012 (2012) · Zbl 1294.62072
[13] Shen, A. T., Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models, Abstract and Applied Analysis, 2013 (2013) · Zbl 1470.62056
[14] Lehmann, E. L., Some concepts of dependence, The Annals of Mathematical Statistics, 37, 5, 1137-1153 (1966) · Zbl 0146.40601
[15] Joag-Dev, K.; Proschan, F., Negative association of random variables, with applications, The Annals of Statistics, 11, 1, 286-295 (1983) · Zbl 0508.62041
[16] Wu, Q. Y., Complete convergence for weighted sums of sequences of negatively dependent random variables, Journal of Probability and Statistics, 2011 (2011) · Zbl 1221.60041
[17] Volodin, A., On the Kolmogorov exponential inequality for negatively dependent random variables, Pakistan Journal of Statistics, 18, 2, 249-253 (2002) · Zbl 1128.60304
[18] Asadian, N.; Fakoor, V.; Bozorgnia, A., Rosenthal’s type inequalities for negatively orthant dependent random variables, Journal of the Iranian Statistical Society, 5, 1-2, 66-75 (2006) · Zbl 1490.60044
[19] Kim, H. C., The Hájeck-Rènyi inequality for weighted sums of negatively orthant dependent random variables, International Journal of Contemporary Mathematical Sciences, 1, 5-8, 297-303 (2006) · Zbl 1156.60306
[20] Amini, M.; Azarnoosh, H. A.; Bozorgnia, A., The strong law of large numbers for negatively dependent generalized Gaussian random variables, Stochastic Analysis and Applications, 22, 4, 893-901 (2004) · Zbl 1056.60024
[21] Amini, M.; Zarei, H.; Bozorgnia, A., Some strong limit theorems of weighted sums for negatively dependent generalized Gaussian random variables, Statistics & Probability Letters, 77, 11, 1106-1110 (2007) · Zbl 1120.60022
[22] Ko, M. H.; Kim, T. S., Almost sure convergence for weighted sums of negatively orthant dependent random variables, Journal of the Korean Mathematical Society, 42, 5, 949-957 (2005) · Zbl 1096.60017
[23] Klesov, O.; Rosalsky, A.; Volodin, A. I., On the almost sure growth rate of sums of lower negatively dependent nonnegative random variables, Statistics & Probability Letters, 71, 2, 193-202 (2005) · Zbl 1070.60030
[24] Wang, X. J.; Hu, S. H.; Shen, A. T.; Yang, W. Z., An exponential inequality for a NOD sequence and a strong law of large numbers, Applied Mathematics Letters, 24, 2, 219-223 (2011) · Zbl 1205.60068
[25] Amini, M.; Bozorgnia, A., Complete convergence for negatively dependent random variables, Journal of Applied Mathematics and Stochastic Analysis, 16, 2, 121-126 (2003) · Zbl 1040.60021
[26] Kuczmaszewska, A., On some conditions for complete convergence for arrays of rowwise negatively dependent random variables, Stochastic Analysis and Applications, 24, 6, 1083-1095 (2006) · Zbl 1108.60021
[27] Taylor, R. L.; Patterson, R. F.; Bozorgnia, A., A strong law of large numbers for arrays of rowwise negatively dependent random variables, Stochastic Analysis and Applications, 20, 3, 643-656 (2002) · Zbl 1003.60032
[28] Zarei, H.; Jabbari, H., Complete convergence of weighted sums under negative dependence, Statistical Papers, 52, 2, 413-418 (2011) · Zbl 1247.60044
[29] Wu, Q. Y., Complete convergence for negatively dependent sequences of random variables, Journal of Inequalities and Applications, 2010 (2010) · Zbl 1202.60050
[30] Sung, S. H., Complete convergence for weighted sums of negatively dependent random variables, Statistical Papers, 53, 1, 73-82 (2012) · Zbl 1314.62096
[31] Wang, X. J.; Hu, S. H.; Yang, W. Z., Complete convergence for arrays of rowwise negatively orthant dependent random variables, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales A, 106, 2, 235-245 (2012) · Zbl 1260.60062
[32] Wang, X. J.; Hu, S. H.; Yang, W. Z.; Ling, N. X., Exponential inequalities and inverse moment for NOD sequence, Statistics & Probability Letters, 80, 5-6, 452-461 (2010) · Zbl 1186.60015
[33] Shen, A. T., Some strong limit theorems for arrays of rowwise negatively orthant-dependent random variables, Journal of Inequalities and Applications, 2011 (2011) · Zbl 1276.60034
[34] Shen, A. T., On the strong convergence rate for weighted sums of arrays of rowwise negatively orthant dependent random variables, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales A, 107, 2, 257-271 (2013) · Zbl 1278.60060
[35] Bozorgnia, A.; Patterson, R. F.; Taylor, R. L., Limit theorems for dependent random variables, Proceeding of the World Congress Nonlinear Analysts (WCNA ’92), de Gruyter · Zbl 0845.60010
[36] Wu, Q. Y., A strong limit theorem for weighted sums of sequences of negatively dependent random variables, Journal of Inequalities and Applications, 2010 (2010) · Zbl 1202.60044
[37] Wu, Q. Y., A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables, Journal of Inequalities and Applications, 2012 (2012) · Zbl 1293.62053
[38] Shen, A. T.; Wu, R. C., Strong and weak convergence for asymptotically almost negatively associated random variables, Discrete Dynamics in Nature and Society, 2013 (2013) · Zbl 1269.60036
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.