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The asymptotic properties of least square estimators in the linear errors-in-variables regression model with φ-mixing errors

  • 1.

    School of Mathematical and Finance, Chuzhou University, Chuzhou 239000, China

  • 2.

    China Electronics Technology Group Corporation No.58 Research Institute, Nanjing 210000, China

  • 3.

    School of Big Data and Artificial Intelligence, Chizhou University, Chizhou 247000, China

Cite this:
https://doi.org/10.52396/JUST-2020-0019
  • Received Date: 09 December 2020
  • Rev Recd Date: 30 January 2021
  • Publish Date: 28 February 2021
    Abstract

  • Abstract

    The simple linear errors-in-variables (EV) model with φ-mixing random errors was mainly studied. By using the central limit theorem and the Marcinkiewicz-type strong law of large numbers for the φ-mixing sequence, the asymptotic normality of the least square (LS) estimators for the unknown parameters were established under some mild conditions. In addition, based on the strong convergence for weighted sums of φ-mixing random variables, the strong consistency of the LS estimators were obtained. Finally, the simulation study was provided to verify the validity of the theoretical results.

    Abstract

    The simple linear errors-in-variables (EV) model with φ-mixing random errors was mainly studied. By using the central limit theorem and the Marcinkiewicz-type strong law of large numbers for the φ-mixing sequence, the asymptotic normality of the least square (LS) estimators for the unknown parameters were established under some mild conditions. In addition, based on the strong convergence for weighted sums of φ-mixing random variables, the strong consistency of the LS estimators were obtained. Finally, the simulation study was provided to verify the validity of the theoretical results.
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    • References(26)
  • [1]
    Mittag H J. Estimating parameters in a simple errors-in-variables model: A new approach based on finite sample distribution theory. Statistical Papers, 1989, 30: 133-140.
    [2]
    Fuller W A. Measurement Error Models. New York: Wiley, 1987.
    [3]
    Liu J X, Chen X R. Consistency of LS estimator in simple linear EV regression models. Acta Mathematica Scientia, Series B, 2005, 25(1): 50-58.
    [4]
    Miao Y, Yang G Y, Shen L M. The central limit theorem for LS estimator in simple linear EV regression models.Communications in Statistics-Theory and Methods, 2007, 36(12): 2263-2272.
    [5]
    Miao Y, Yang G Y. The loglog law for LS estimator in simple linear EV regression models. Statistics, 2011, 45(2): 155-162.
    [6]
    Miao Y, Wang K, Zhao F F. Some limit behaviors for the LS estimator in simple linear EV regression models. Statistics and Probability Letters, 2011, 81(1): 92-102.
    [7]
    Fan G L, Liang H Y, Wang J F, et al. Asymptotic properties for LS estimators in EV regression model with dependent errors. AStA Advances in Statistical Analysis, 2010, 94: 89-103.
    [8]
    Yang Q L. Asymptotic normality of LS estimators in the simple linear EV regression model with PA errors. Communications in Statistics-Theory and Methods, 2012, 41(23): 4276-4284.
    [9]
    Miao Y, Zhao F F, Wang K, et al. Asymptotic normality and strong consistency of LS estimators in the EV regression model with NA errors. Statistical Papers, 2013, 54: 193-206.
    [10]
    Wang X J, Shen A T, Chen Z Y, et al. Complete convergence for weighted sums of NSD random variables and its application in the EV regression model. Test, 2015, 24: 166-184.
    [11]
    Wang X J, Wu Y, Hu S H. Strong and weak consistency of LS estimators in the EV regression model with negatively superadditive-dependent errors. AStA Advances in Statistical Analysis, 2018, 102: 41-65.
    [12]
    Wang X J, Xi M M, Wang H X, et al. On consistency of LS estimators in the errors-in-variable regression model. Probability in the Engineering and Informational Sciences, 2018, 32: 144-162.
    [13]
    Shen A T. Asymptotic properties of LS estimators in the errors-in-variables model with MD errors. Statistical Papers, 2019, 60(4): 1193-1206.
    [14]
    Dobrushin R L. The central limit theorem for non-stationary Markov chain. Theory of Probability and Its Applications, 1956, 1: 72-88.
    [15]
    Babu G J, Ghosh M, Singh K. On rates of convergence to normality for φ-mixing processes. Sankhya, Series A, 1978, 40(3): 278-293.
    [16]
    Utev S A. The central limit theorem for φ-mixing arrays of random variables. Theory of Probability and Its Applications, 1990, 35(1): 131-139.
    [17]
    Kiesel R. Strong laws and summability for φ-mixing sequences of random variables. Journal of Theoretical Probability, 1998, 11(1): 209-224.
    [18]
    Hu S H, Wang X J. Large deviations for some dependent sequences. Acta Mathematica Scientic, Series B, 2008, 28(2): 295-300.
    [19]
    Yang W Z, Wang X J, Li X Q, et al. Berry-Esseen bound of sample quantiles for φ-mixing random variables. Journal of Mathematical Analysis and Applications, 2012, 388: 451-462.
    [20]
    Shen A T, Wang X H, Ling J M. On complete convergence for non-stationary φ-mixing random variables. Communications in Statistics-Theory and Methods, 2014, 43(22): 4856-4866.
    [21]
    Billingsley P. Convergence of Probability Measures. New York : Wiley, 1968.
    [22]
    Lu C R, Lin Z Y. Limit Theory for Mixing Dependent Sequences. Beijing: Science Press of China, 1997.
    [23]
    Peligrad M, Utev S. Central limit theorem for linear processes. The Annals of Probability, 1997, 25(1): 443-456.
    [24]
    Thanh Lv, Yin G. Weighted sums of strongly mixing random variables with an application to nonparametric regression. Statistics and Probability Letters, 2015, 105: 195-202.
    [25]
    Wang X J, Hu S H. Some Baum-Katz type results for φ-mixing random variables with different distributions. RACSAM, 2012, 106: 321-331.
    [26]
    Wu Q Y. Further study strong consistency of M estimator in linear model for ρ-mixing random samples. Journal of Systems Science and Complexity, 2011, 24: 969-980.
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    [1]
    Mittag H J. Estimating parameters in a simple errors-in-variables model: A new approach based on finite sample distribution theory. Statistical Papers, 1989, 30: 133-140.
    [2]
    Fuller W A. Measurement Error Models. New York: Wiley, 1987.
    [3]
    Liu J X, Chen X R. Consistency of LS estimator in simple linear EV regression models. Acta Mathematica Scientia, Series B, 2005, 25(1): 50-58.
    [4]
    Miao Y, Yang G Y, Shen L M. The central limit theorem for LS estimator in simple linear EV regression models.Communications in Statistics-Theory and Methods, 2007, 36(12): 2263-2272.
    [5]
    Miao Y, Yang G Y. The loglog law for LS estimator in simple linear EV regression models. Statistics, 2011, 45(2): 155-162.
    [6]
    Miao Y, Wang K, Zhao F F. Some limit behaviors for the LS estimator in simple linear EV regression models. Statistics and Probability Letters, 2011, 81(1): 92-102.
    [7]
    Fan G L, Liang H Y, Wang J F, et al. Asymptotic properties for LS estimators in EV regression model with dependent errors. AStA Advances in Statistical Analysis, 2010, 94: 89-103.
    [8]
    Yang Q L. Asymptotic normality of LS estimators in the simple linear EV regression model with PA errors. Communications in Statistics-Theory and Methods, 2012, 41(23): 4276-4284.
    [9]
    Miao Y, Zhao F F, Wang K, et al. Asymptotic normality and strong consistency of LS estimators in the EV regression model with NA errors. Statistical Papers, 2013, 54: 193-206.
    [10]
    Wang X J, Shen A T, Chen Z Y, et al. Complete convergence for weighted sums of NSD random variables and its application in the EV regression model. Test, 2015, 24: 166-184.
    [11]
    Wang X J, Wu Y, Hu S H. Strong and weak consistency of LS estimators in the EV regression model with negatively superadditive-dependent errors. AStA Advances in Statistical Analysis, 2018, 102: 41-65.
    [12]
    Wang X J, Xi M M, Wang H X, et al. On consistency of LS estimators in the errors-in-variable regression model. Probability in the Engineering and Informational Sciences, 2018, 32: 144-162.
    [13]
    Shen A T. Asymptotic properties of LS estimators in the errors-in-variables model with MD errors. Statistical Papers, 2019, 60(4): 1193-1206.
    [14]
    Dobrushin R L. The central limit theorem for non-stationary Markov chain. Theory of Probability and Its Applications, 1956, 1: 72-88.
    [15]
    Babu G J, Ghosh M, Singh K. On rates of convergence to normality for φ-mixing processes. Sankhya, Series A, 1978, 40(3): 278-293.
    [16]
    Utev S A. The central limit theorem for φ-mixing arrays of random variables. Theory of Probability and Its Applications, 1990, 35(1): 131-139.
    [17]
    Kiesel R. Strong laws and summability for φ-mixing sequences of random variables. Journal of Theoretical Probability, 1998, 11(1): 209-224.
    [18]
    Hu S H, Wang X J. Large deviations for some dependent sequences. Acta Mathematica Scientic, Series B, 2008, 28(2): 295-300.
    [19]
    Yang W Z, Wang X J, Li X Q, et al. Berry-Esseen bound of sample quantiles for φ-mixing random variables. Journal of Mathematical Analysis and Applications, 2012, 388: 451-462.
    [20]
    Shen A T, Wang X H, Ling J M. On complete convergence for non-stationary φ-mixing random variables. Communications in Statistics-Theory and Methods, 2014, 43(22): 4856-4866.
    [21]
    Billingsley P. Convergence of Probability Measures. New York : Wiley, 1968.
    [22]
    Lu C R, Lin Z Y. Limit Theory for Mixing Dependent Sequences. Beijing: Science Press of China, 1997.
    [23]
    Peligrad M, Utev S. Central limit theorem for linear processes. The Annals of Probability, 1997, 25(1): 443-456.
    [24]
    Thanh Lv, Yin G. Weighted sums of strongly mixing random variables with an application to nonparametric regression. Statistics and Probability Letters, 2015, 105: 195-202.
    [25]
    Wang X J, Hu S H. Some Baum-Katz type results for φ-mixing random variables with different distributions. RACSAM, 2012, 106: 321-331.
    [26]
    Wu Q Y. Further study strong consistency of M estimator in linear model for ρ-mixing random samples. Journal of Systems Science and Complexity, 2011, 24: 969-980.
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