Complete $f$-moment convergence for weighted sums of WOD arrays with statistical applications

Chen, Xi ; Tao, Xinran ; Wang, Xuejun

Kybernetika, Tome 59 (2023) no. 1, pp. 1-27

Abstract (VO)

Complete $f$-moment convergence is much more general than complete convergence and complete moment convergence. In this work, we mainly investigate the complete $f$-moment convergence for weighted sums of widely orthant dependent (WOD, for short) arrays. A general result on Complete $f$-moment convergence is obtained under some suitable conditions, which generalizes the corresponding one in the literature. As an application, we establish the complete consistency for the weighted linear estimator in nonparametric regression models. Finally, some simulations are provided to show the numerical performance of theoretical results based on finite samples.

Détail

DOI : 10.14736/kyb-2023-1-0001

Classification : 60F15, 62G20
Keywords: widely orthant dependent arrays; weighted sums; complete $f$-moment convergence; complete convergence; nonparametric regression models; complete consistency

Bibliographie

[1] Adler, A., Rosalsky, A.: Some general strong laws for weighted sums of stochastically dominated random variables. Stoch. Anal. Appl. 5 (1987), 1-16. | DOI | MR | Zbl

[2] Adler, A., Rosalsky, A., Taylor, R. L.: Strong laws of large numbers for weighted sums of random elements in normed linear spaces. Int. J. Math. Math. Sci. 12 (1989), 507-530. | DOI | MR

[3] Chen, P. Y., Sung, S. H.: A Spitzer-type law of large numbers for widely orthant dependent random variables. Statist. Probab. Lett. 2054 (2019), 1-8, Article ID 108544. | DOI | MR

[4] Chow, Y. S.: On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math., Academia Sinica 16 (1988), 177-201. | MR

[5] Georgiev, A. A.: Local properties of function fitting estimates with applications to system identification. In: Mathematics Statistics and Applications. Proceedings 4th Pannonian Symposium on Mathematical Statistics 1983, (W. Grossmann, ed.). vol. B, Bad Tatzmannsdorf, Austria, Reidel, Dordrecht, pp. 141-51. | MR

[6] Georgiev, A. A., Greblicki, W.: Nonparametric function recovering from noisy observations. J. Statist. Plann. Inference 13 (1986), 1-14. | DOI | MR

[7] He, Q. H.: Consistency of the Priestley-Chao estimator in nonparametric regression model with widely orthant dependent errors. J. Inequal. Appl. 2019 (2019), 1-13, Article ID 64. | DOI | MR

[8] Hsu, P. L., Robbins, H.: Complete convergence and the law of large numbers. Proc. National Acad. Sci. Unit. States Amer. 33 (1947), 25-31. | DOI | MR | Zbl

[9] Joag-Dev, K., Proschan, F.: Negative association of random variables with applications. Ann. Statist. 11 (1983), 286-295. | DOI | MR

[10] Lang, J. J., He, T. Y., Cheng, L., Lu, C., Wang, X. J.: Complete convergence for weighted sums of widely orthant-dependent random variables and its statistical application. Revista Mat. Complut. 34 (2021), 853-881. | DOI | MR

[11] Li, Y. M., Zhou, Y., Liu, C.: On the convergence rates of kernel estimator and hazard estimator for widely dependent samples. J. Inequal. Appl. 2018 (2018), 1-10, Article ID 71. | DOI | MR

[12] Liang, H. Y., Jing, B. Y.: Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences. J. Multivar. Anal. 95 (2005), 227-245. | DOI | MR

[13] Liang, H. Y., Zhang, J. J.: Strong convergence for weighted sums of negatively associated arrays. Chinese Annals of Mathematics 31B (2010), 273-288. | DOI | MR

[14] Lu, C., Chen, Z., Wang, X. J.: Complete $f$-moment convergence for widely orthant dependent random variables and its application in nonparametric models. Acta Math. Sinica, English Series 35 (2019), 1917-1936. | DOI | MR

[15] Hu, D. H. Qiu abd T. C.: Strong limit theorems for weighted sums of widely orthant dependent random variables. J. Math. Res. Appl. 34 (2014), 105-113. | MR

[16] Qiu, D. H., Chen, P. Y.: Complete and complete moment convergence for weighted sums of widely orthant dependent random variables. Acta Math. Sinica, English Series 30 (2014), 1539-1548. | DOI | MR

[17] Roussas, G. G., Tran, L. T., Ioannides, D. A.: Fixed design regression for time series: asymptotic. J. Multivar. Anal. 40 (1992), 262-291. | DOI | MR

[18] Shen, A.T.: Complete convergence for weighted sums of END random variables and its application to nonparametric regression models. J. Nonparametr. Statist. 28 (2016), 702-715. | DOI | MR

[19] Shen, A. T., Wu, C. Q.: Complete $q$th moment convergence and its statistical applications. RACSAM 114 (2019), 1-25, Article ID 35. | MR

[20] Shen, A. T., Yao, M., Wang, W. J., Volodin, A.: Exponential probability inequalities for WNOD random variables and their applications. RACSAM 110 (2016), 251-268. | DOI | MR

[21] Shen, A. T., Zhang, S. Y.: On complete consistency for the estimator of nonparametric regression model based on asymptotically almost negatively associated errors. Methodol. Comput. Appl. Probab. 23 (2021), 1285-1307. | DOI | MR

[22] Shen, A. T., Zhang, Y., Volodin, A.: Applications of the Rosenthal-type inequality for negatively super-additive dependent random variables. Metrika 78 (2015), 295-311. | DOI | MR

[23] Stout, W. F.: Almost Sure Convergence. Academic Press, New York 1974. | MR

[24] Stone, C. J.: Consistent nonparametric regression regression. Ann. Statist. 5 (1977), 595-620. | DOI | MR

[25] Tran, L., Roussas, G., Yakowitz, S., Truong, V. B.: Fixed design regression for linear time series. Ann. Statist. 24 (1996), 975-991. | DOI | MR

[26] Wang, Y., Wang, X. J.: Complete $f$-moment convergence for Sung's type weighted sums and its application to the EV regression models. Statist. Papers 62 (2021), 769-793. | DOI | MR

[27] Wang, K. Y., Wang, Y. B., Gao, Q. W.: Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate. Methodol. Comput. Appl. Probab. 15 (2013), 109-124. | DOI | MR

[28] Xi, M. M., Wang, R., Cheng, Z. Y., Wang, X. J.: Some convergence properties for partial sums of widely orthant dependent random variables and their statistical applications. Statist. Papers 61 (2020), 1663-1684. | DOI | MR

[29] Wu, Q. Y.: Probability Limit Theory for Mixing Sequences. Science Press of China, Beijing 2006.

[30] Wu, Y., Wang, X. J., Hu, S. H.: Complete moment convergence for weighted sums of weakly dependent random variables and its application in nonparametric regression model. Statist. Probab. Lett. 127 (2017), 56-66. | DOI | MR

[31] Wu, Y., Wang, X. J., Hu, T. C., Volodin, A.: Complete $f$-moment convergence for extended negatively dependent random variables. RACSAM 113 (2019), 333-351. | DOI | MR

[32] Wu, Y., Wang, X. J., Rosalsky, A.: Complete moment convergence for arrays of rowwise widely orthant dependent random variables. Acta Math. Sinica, English Series 34 (2018), 1531-1548. | DOI | MR

[33] Wu, Y., Wang, X. J., Shen, A. T.: Strong convergence properties for weighted sums of m-asymptotic negatively associated random variables and statistical applications. Statist. Papers 62 (2021), 2169-94. | DOI | MR

[34] Yang, W. Z., Xu, H. Y., Chen, L., Hu, aand S. H.: Complete consistency of estimators for regression models based on extended negatively dependent. Statist. Papers 59 (2018), 449-465. | DOI | MR

[35] Zhang, S. L., Qu, C., Hou, T. T.: Limit behaviors of the estimator of nonpar ametric regression model based on extended negatively dependent errors. Commun. Statist. - Theory Methods, 2022, in press. | DOI | MR

[36] Zhou, X. C., Lin, J. G., Yin, C. M.: Asymptotic properties of wavelet-based estimator in nonparametric regression model with weakly dependent processes. J. Inequal. Appl. 2013 (2013), 1-18, Article ID 261. | DOI | MR

[37] Wang, X. J., Xu, C., Hu, T. C., Volodin, A., Hu, S. H.: On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST 23 (2014), 607-629. | DOI | MR