Complete \(f\)-moment convergence for arrays of rowwise \(m\)-negatively associated random variables and its statistical applications. (English) Zbl 07725440
Summary: In this paper, we study the complete \(f\)-moment convergence for arrays of rowwise \(m\)-negatively associated random variables under some general conditions. The results obtained in the paper extend and improve some previous known ones. As an application of the main results, we present the complete consistency for the estimator in a semiparametric regression model based on \(m\)-negatively associated errors. We perform some numerical simulations to verify the validity of the theoretical results based on finite samples.
MSC:
| 62G20 | Asymptotic properties of nonparametric inference |
Keywords:
complete consistency; complete convergence; complete \(f\)-moment convergence; \(m\)-negatively associated random variables; semiparametric regression modelReferences:
| [1] | Adler, A.; Rosalsky, A., Some general strong laws for weighted sums of stochastically dominated random variables, Stoch. Anal. Appl, 5, 1-16 (1987) · Zbl 0617.60028 · doi:10.1080/07362998708809104 |
| [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, 507-529 (1989) · Zbl 0679.60009 · doi:10.1155/S0161171289000657 |
| [3] | Arashi, M.; Roozbeh, M., Some improved estimation strategies in high-dimensional semiparametric regression models with application to riboflavin production data, Stat. Pap, 60, 667-686 (2019) · Zbl 1419.62079 · doi:10.1007/s00362-016-0843-y |
| [4] | Baek, J. I.; Choi, I. B.; Niu, S. L., On the complete convergence of weighted sums for arrays of negatively associated variables, J. Korean Stat. Soc, 37, 73-80 (2008) · Zbl 1298.60039 · doi:10.1016/j.jkss.2007.08.001 |
| [5] | Baek, J. I.; Liang, H. Y., Asymptotics of estimators in semi-parametric model under NA samples, J. Stat. Plan. Inference, 136, 3362-3382 (2006) · Zbl 1112.62032 · doi:10.1016/j.jspi.2005.01.008 |
| [6] | Bravo, F., Semiparametric quantile regression with random censoring, Ann. Inst. Stat. Math, 72, 265-295 (2020) · Zbl 1436.62129 · doi:10.1007/s10463-018-0688-3 |
| [7] | Budsaba, K.; Chen, P.; Panishkan, K.; Volodin, A., Strong laws for certain types of U-statistics based on negatively associated random variables, Sib. Adv. Math, 19, 225-232 (2009) · doi:10.3103/S1055134409040014 |
| [8] | Cai, G. H., Strong laws for weighted sums of NA random variables, Metrika, 68, 323-331 (2008) · Zbl 1247.60036 · doi:10.1007/s00184-007-0160-5 |
| [9] | Chaubey, Y. P.; Doosti, H.; Prakasa Rao, B. L.S.., Wavelet based estimation of the derivatives of a density for a negatively associated process, J. Stat. Theory Pract, 2, 453-463 (2008) · Zbl 1211.62061 · doi:10.1080/15598608.2008.10411886 |
| [10] | Chen, P.; Chen, P.; Hu, T.-C.; Liu, X. J.; Volodin, A. I., On complete convergence for arrays of rowwise negatively associated random variables, Theory Probab. Appl, 52, 393-397 (2007) · Zbl 1146.60025 · doi:10.4213/tvp183 |
| [11] | Chen, P. Y.; Sung, S. H., On the strong convergence for weighted sums of negatively associated random variables, Stat. Probab. Lett, 92, 45-52 (2014) · Zbl 1291.60055 · doi:10.1016/j.spl.2014.04.028 |
| [12] | Choongrak, K.; Byeong, U. P.; Woochul, K., Influence diagnostics in semiparametric regression models, Stat. Probab. Lett, 60, 49-58 (2002) · Zbl 1092.62525 |
| [13] | Chow, Y. S., On the rate of moment convergence of sample sums and extremes, Bull. Inst. Math. Acad. Sin, 16, 177-201 (1988) · Zbl 0655.60028 |
| [14] | Emmanuel, D. N.; Guy, M. N., Strong consistency of kernel estimator in a semiparametric regression model, Statistics, 53, 1289-1305 (2019) · Zbl 1437.62246 |
| [15] | Fu, K. A.; Zhang, L. X., Precise rates in the law of the logarithm for negatively associated random variable, Comput. Math. Appl, 54, 687-698 (2007) · Zbl 1155.60314 · doi:10.1016/j.camwa.2007.02.008 |
| [16] | Gan, S. X.; Chen, P. Y., On the limiting behavior of the maximum partial sums for arrays of rowwise NA random variables, Acta Math. Sci. Ser. B, 27, 283-290 (2007) · Zbl 1125.60027 · doi:10.1016/S0252-9602(07)60027-7 |
| [17] | Guo, H. J.; Kou, J. K., Wavelet regression estimations for negatively associated sample, Appl. Anal, 99, 2657-2669 (2020) · Zbl 1476.42022 · doi:10.1080/00036811.2019.1577392 |
| [18] | Hien, N. T.T.; Thanh, L. V., On the weak laws of large numbers for sums of negatively associated random vectors in Hilbert spaces, Stat. Probab. Lett, 107, 236-245 (2015) · Zbl 1329.60029 · doi:10.1016/j.spl.2015.08.030 |
| [19] | Hsu, P. L.; Robbins, H., Complete convergence and the law of large numbers, Proc. Natl. Acad. Sci. USA, 33, 25-31 (1947) · Zbl 0030.20101 · doi:10.1073/pnas.33.2.25 |
| [20] | Hu, H. C.; Hu, D. H., Strong consistency of wavelet estimation in semiparametric regression models, Acta Math. Sin. Chin. Ser, 49, 1417-1424 (2006) · Zbl 1107.62034 |
| [21] | Hu, S. H., Estimate for a semiparametric regression model, Acta Math. Sci. Ser. A, 19, 541-549 (1999) · Zbl 0954.62050 |
| [22] | Hu, S. H., Fixed-design semiparametric regression for linear time series, Acta Math. Sci. Ser. B, 26, 74-82 (2006) · Zbl 1092.62047 · doi:10.1016/S0252-9602(06)60028-3 |
| [23] | Hu, T. C.; Chiang, C. Y.; Taylor, R. L., On complete convergence for arrays of rowwise m-negatively associated random variables, Nonlinear Anal. Theory Methods Appl, 71, 1075-1081 (2009) · Zbl 1238.60041 |
| [24] | Hu, Y. J.; Ming, R. X.; Yang, W. Q., Large deviations and moderate deviations for m-negatively associated random variables, Acta Math. Sci. Ser. B, 27, 886-896 (2007) · Zbl 1150.60010 |
| [25] | Huan, N. V.; Quang, N. V.; Thuan, N. T., Baum-Katz type theorems for coordinatewise negatively associated random vectors in Hilbert spaces, Acta Math. Hungar, 144, 132-149 (2014) · Zbl 1349.60046 · doi:10.1007/s10474-014-0424-2 |
| [26] | Joag-Dev, K.; Proschan, F., Negative association of random variables with applications, Ann. Stat, 11, 286-295 (1983) · Zbl 0508.62041 · doi:10.1214/aos/1176346079 |
| [27] | Kazembe, L. N., Modelling individual fertility levels in malawian women: a spatial semiparametric regression model, Stat. Methods Appl, 18, 237-255 (2009) · Zbl 1405.62240 · doi:10.1007/s10260-007-0076-2 |
| [28] | Kuczmaszewska, A., On complete convergence for arrays of rowwise negatively associated random variables, Stat. Probab. Lett, 79, 116-124 (2009) · Zbl 1154.60319 · doi:10.1016/j.spl.2008.07.030 |
| [29] | 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. Sin. English Ser, 35, 1917-1936 (2019) · Zbl 1428.60050 · doi:10.1007/s10114-019-8315-7 |
| [30] | Matuła, P., A note on the almost sure convergence of sums of negatively dependent random variables, Stat. Probab. Lett, 15, 209-213 (1992) · Zbl 0925.60024 · doi:10.1016/0167-7152(92)90191-7 |
| [31] | Mikusheva, A. E., On the complete convergence of sums of negatively associated random variables, Math. Notes, 68, 355-362 (2000) · Zbl 0988.60051 · doi:10.1007/BF02674559 |
| [32] | Nooghabi, H. J.; Azarnoosh, H. A., Exponential inequality for negatively associated random variables, Stat. Pap, 50, 419-428 (2009) · Zbl 1247.60039 · doi:10.1007/s00362-007-0081-4 |
| [33] | Pan, G. M.; Hu, S. H.; Fang, L. B.; Cheng, Z. D., Mean consistency for a semiparametric regression model, Acta Math. Sci. Ser. A, 23, 598-606 (2003) · Zbl 1052.62043 |
| [34] | Qin, Y. S.; Li, Y. H., Empirical likelihood for linear models under negatively associated errors, J. Syst. Sci. Complex, 102, 153-163 (2011) · Zbl 1206.62055 · doi:10.1016/j.jmva.2010.08.010 |
| [35] | Ren, Z.; Chen, M. H., Strong consistency of a class of estimators in semiparametric regression model when the sample is a sequence of NA, Appl. Math. A: J. Chin. Univ. Ser. A, 4, 467-474 (2000) · Zbl 1086.62501 |
| [36] | Shao, Q. M., A comparison theorem on moment inequalities between negatively associated and independent random variables, J. Theor. Probab, 13, 343-356 (2000) · Zbl 0971.60015 · doi:10.1023/A:1007849609234 |
| [37] | Shen, A. T.; Zhang, Y.; Xiao, B. Q.; Volodin, A., Moment inequalities for m-negatively associated random variables and their applications, Stat. Pap, 58, 911-928 (2015) · Zbl 1387.60058 |
| [38] | Su, C.; Zhao, L. X.; Wang, Y. B., Moment inequalities and weak convergence for negatively associated sequences, Sci. China Ser. A: Math, 40, 172-182 (1997) · Zbl 0907.60023 · doi:10.1007/BF02874436 |
| [39] | Sung, S. H.; Srisuradetchai, P.; Volodin, A., A note on the exponential inequality for a class of dependent random variables, J. Korean Stat. Soc, 40, 109-114 (2011) · Zbl 1296.60047 · doi:10.1016/j.jkss.2010.08.002 |
| [40] | Wang, M. G.; Wang, X. J., Some convergence properties for the maximum of partial sums of m-negatively associated random variables, RACSAM, 113, 2345-2358 (2019) · Zbl 1448.60075 |
| [41] | Wang, X. J.; Ge, M. M.; Wu, Y., The asymptotic properties of the estimators in a semiparametric regression model, Stat. Pap, 60, 2087-2108 (2019) · Zbl 1432.62082 · doi:10.1007/s00362-017-0910-z |
| [42] | Wang, X. J.; Wu, Y.; Hu, S. H.; Ling, N. X., Complete moment convergence for negatively orthant dependent random variables and its application in statistical models, Stat. Pap, 61, 1147-1180 (2020) · Zbl 1447.60057 · doi:10.1007/s00362-018-0983-3 |
| [43] | Wang, Y.; Li, X. Q.; Chen, L.; Wang, M. H.; Wang, X. J., A note on the asymptotic properties of the estimators in a semiparametric regression model, Commun. Stat.—Simul. Comput, 51, 358-373 (2022) · Zbl 1524.62195 · doi:10.1080/03610918.2019.1652316 |
| [44] | Wang, Y.; Wang, X. J., Complete f-moment convergence for Sung’s type weighted sums and its application to the EV regression models, Stat. Pap, 62, 769-793 (2021) · Zbl 1482.60049 · doi:10.1007/s00362-019-01112-z |
| [45] | Wiroonsri, N., Stein’s method for negatively associated random variables with applications to second-order stationary random fields, J. Appl. Probab, 55, 196-215 (2018) · Zbl 1401.60040 · doi:10.1017/jpr.2018.13 |
| [46] | Wu, Y.; Wang, X. J.; Hu, T. C.; Volodin, A., Complete f-moment convergence for extended negatively dependent random variables, RACSAM, 113, 333-351 (2019) · Zbl 1415.60030 · doi:10.1007/s13398-017-0480-x |
| [47] | Wu, Y. F., Convergence properties of the maximal partial sums for arrays of rowwise NA random variables, Theory Probab. Appl, 56, 527-535 (2012) · Zbl 1267.60035 |
| [48] | Wu, Y. F.; Hu, T. C.; Volodin, A., Complete convergence and complete moment convergence for weighted sums of m-NA random variables, J. Inequal. Appl, 2015, 200 (2015) · Zbl 1379.60034 |
| [49] | Xing, G. D., On the exponential inequalities for strictly stationary and negatively associated random variables, J. Stat. Plan. Inference, 139, 3453-3460 (2009) · Zbl 1181.60042 · doi:10.1016/j.jspi.2009.03.023 |
| [50] | Yang, S. J., Estimation for semiparametric varying coefficient models with different smoothing variables under random right censoring, J. Korean Stat. Soc, 47, 161-171 (2018) · Zbl 1390.62069 · doi:10.1016/j.jkss.2017.12.001 |
| [51] | Zhou, X. C.; Lin, J. G., Asymptotic properties of wavelet estimators in semiparametric regression models under dependent errors, J. Multivar. Anal, 122, 251-270 (2013) · Zbl 1280.62051 · doi:10.1016/j.jmva.2013.08.006 |
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.
