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Articles

On the rates of asymptotic normality for recursive kernel density estimators under ϕ-mixing assumptions

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Pages 340-363 | Received 14 Mar 2018, Accepted 30 Dec 2018, Published online: 16 Jan 2019

References

  • Billingsley, P. (1968), Convergence of Probability Measures, New York: Wiley.
  • Dobrusin, H.L. (1956), ‘Central Limit Theorem for Non-stationary Markov Chains’, Theory of Probability and Its Applications, 1(1), 365–425.
  • Hang, Y., and Xu, X.J. (2016), ‘Analysis of Time Series Model Based on Daily Return Rate of Shanghai Stock Market’, Reform and Opening Up, 2016(5), 21–23.
  • Li, Y.M., Yin, C.M., and Wei, C.D. (2008), ‘On the Asymptotic Normality for ϕ-mixing Dependent Errors of Wavelet Regression Function Estimator’, Acta Mathematicae Applicatae Sinica, 31(6), 1046–1055.
  • Liang, H.Y., and Baek, J.I. (2004), ‘Asymptotic Normality of Recursive Density Estimates Under Some Dependence Assumptions’, Metrika, 60(2), 155–166. doi: 10.1007/s001840300302
  • Liang, H.Y., and Baek, J.I. (2008), ‘Berry-Esseen Bounds for Density Estimates under NA Assumption’, Metrika, 68(3), 305–322. doi: 10.1007/s00184-007-0159-y
  • Liang, H.Y., and Fan, G.L. (2009), ‘Berry-Esseen Type Bounds of Estimators in a Semiparametric Model with Linear Process Errors’, Journal of Multivariate Analysis, 100(1), 1–15. doi: 10.1016/j.jmva.2008.03.006
  • Lin, Z.Y., and Lu, C.R. (1996), Limit Theory for Mixing Dependent Random Variables, New York: Springer.
  • Liu, Y.X., and Niu, S.L. (2016), ‘Berry-Esseen Bounds of Recursive Kernel Estimator of Density Under Strong Mixing Assumptions’, Bulletin of the Korean Mathematical Society, 54(1), 343–358. doi: 10.4134/BKMS.b160139
  • Masry, E. (1987), ‘Almost Sure Convergence of Recursive Density Estimators for Stationary Mixing Processes’, Statistics and Probability Letters, 5(4), 249–254. doi: 10.1016/0167-7152(87)90100-3
  • Masry, E., and Györfi, L. (1987), ‘Strong Consistency and Rates for Recursive Probability Density Estimators of Stationary Processes’, Journal of Multivariate Analysis, 22(1), 79–93. doi: 10.1016/0047-259X(87)90077-7
  • Mokkadem, A., Pelletier, M., and Slaoui, Y. (2009), ‘The Stochastic Approximation Method for the Estimation of a Multivariate Probability Density’, Journal of Statistical Planning and Inference, 139(7), 2459–2478. doi: 10.1016/j.jspi.2008.11.012
  • Shen, A.T., Wang, X.H., Li, X.Q., and Wang, X.J. (2014), ‘On the Rate of Complete Convergence for Weighted Sums of Arrays of Rowwise ϕ-mixing Random Variables’, Communications in Statistics - Theory and Methods, 43(13), 2714–2725. doi: 10.1080/03610926.2012.683130
  • Shen, A.T., Wang, X.H., and Ling, J.M. (2014), ‘On Complete Convergence for Nonstationary ϕ-mixing Random Variables’, Communications in Statistics - Theory and Methods, 43(22), 4856–4866. doi: 10.1080/03610926.2012.725501
  • Shen, A.T. (2014), ‘On Asymptotic Approximation of Inverse Moments for a Class of Nonnegative Random Variables’, Statistics, 48(6), 1371–1379. doi: 10.1080/02331888.2013.801480
  • Slaoui, Y. (2014), ‘Bandwidth Selection for Recursive Kernel Density Estimators Defined by Stochastic Approximation Method’, Journal of Probability and Statistics, 2014, Article ID 739640. doi:10.1155/2014/739640
  • Stout, W.F. (1974), Almost Sure Convergence, New York: Academic Press.
  • Takahata, H. (1980), ‘Almost Sure Convergence of Density Estimators for Weakly Dependent Stationary Processes’, Bulletin of Tokyo Gakugei University, 32, 11–32.
  • Wang, L., and Liang, H.Y. (2004), ‘Strong Uniform Convergence of the Recursive Regression Estimator under ϕ-mixing Conditions’, Metrika, 59(3), 245–261. doi: 10.1007/s001840300282
  • Wang, X.J., Hu, S.H, Shen, Y., and Yang, W.Z. (2009), ‘Moment Inequality for ϕ-mixing Sequences and Its Applications’, Journal of Inequalities and Applications, 2009, Article ID 379743, 12 pages.
  • Wang, X.J., Ge, M.M., and Wu, Y. (2017), ‘The Asymptotic Properties of the Estimators in a Semiparametric Regression Model’, Statistical Papers, doi:10.1007/s00362-017-0910-z
  • Wang, Z.X., and Huang, Y.H. (2018), ‘Analysis of Banking Stock Returns’, Statistics and Application, 7(4), 463–475. doi: 10.12677/SA.2018.74054
  • Wegman, E.J., and Davies, H.I. (1979), ‘Remarks on Some Recursive Estimators of a Probability Density’, The Annals of Statistics, 7(2), 316–327. doi: 10.1214/aos/1176344616
  • Wolverton, C., and Wagner, T. (1969), ‘Asymptotically Optimal Discriminant Functions for Pattern Classification’, IEEE Transactions on Information Theory, 15(2), 258–265. doi: 10.1109/TIT.1969.1054295
  • Yamato, H. (1971), ‘Sequential Estimation of a Continuous Probability Density-function and Mode’, Bulletin of Mathematical Statistics, 14(3–4), 1–12.
  • Yang, S.C. (1995), ‘Almost Sure Convergence of Weighted Sums of Mixing Sequences’, Journal of System Science and Mathematical Sciences, 15(3), 254–265.
  • Yang, W.Z., and Hu, S.H. (2012), ‘The Berry-Esséen Bounds for Kernel Density Estimator Under Dependent Sample, Journal of Inequalities and Applications, 2012, Article ID 287, 13 pages.
  • Yang, W.Z., Wang, X.J., Li, X.Q., and Hu, S.H. (2012), ‘Berry-Esséen Bound of Sample Quantiles for ϕ-mixing Random Variables’, Journal of Mathematical Analysis and Applications, 388(1), 451–462. doi: 10.1016/j.jmaa.2011.10.058

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