Estimation of Covariance Matrix with ARMA Structure Through Quadratic Loss Function

  • Defei Zhang
  • Xiangzhao Cui
  • Chun Li
  • Jianxin PanEmail author


In this paper we propose a novel method to estimate the high-dimensional covariance matrix with an order-1 autoregressive moving average process, i.e. ARMA(1,1), through quadratic loss function. The ARMA(1,1) structure is a commonly used covariance structures in time series and multivariate analysis but involves unknown parameters including the variance and two correlation coefficients. We propose to use the quadratic loss function to measure the discrepancy between a given covariance matrix, such as the sample covariance matrix, and the underlying covariance matrix with ARMA(1,1) structure, so that the parameter estimates can be obtained by minimizing the discrepancy. Simulation studies and real data analysis show that the proposed method works well in estimating the covariance matrix with ARMA(1,1) structure even if the dimension is very high.


ARMA(1 1) structure Covariance matrix Quadratic loss function 



This research is supported by the National Science Foundation of China (11761028 and 11871357). We acknowledge helpful comments and insightful suggestions made by a referee.


  1. 1.
    Fan, J., Fan, Y., Lv, J.: High dimensional covariance matrix estimation using a factor model. J. Econ. 147(1), 186–197 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Francq, C.: Covariance matrix estimation for estimators of mixing weak ARMA models. J. Stat. Plan. Inference 83(2), 369–394 (2000)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Haff, L.R.: Empirical bayes estimation of the multivariate normal covariance matrix. Ann. Statist. 8(3), 586–597 (1980)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge, UK (2013)zbMATHGoogle Scholar
  5. 5.
    Kenward M. G.: A method for comparing profiles of repeated measurements. Applied Statistics, 36(3), 296–308 (1987)MathSciNetGoogle Scholar
  6. 6.
    Lin, F. Jovanovi\(\acute{c}\), M.R.: Least-squares approximation of structured covariances, IEEE Trans. Automat. Control 54(7), 1643–1648 (2009)Google Scholar
  7. 7.
    Lin, L., Higham, N.J., Pan, J.: Covariance structure regularization via entropy loss function. Comput. Stat. Data Anal. 72(4), 315–327 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ning, L., Jiang, X., Georgiou, T.: Geometric methods for structured covariance estimation. In: American Control Conference, pp. 1877–1882. IEEE (2012)Google Scholar
  9. 9.
    Olkin, I., Selliah, J.B.: Estimating covariance matrix in a multivariate normal distribution. In: Gupta, S.S., Moore, D.S. (eds.) Statistical Decision Theory and Related Topics, vol. II, pp. 313–326. Academic Press, New York (1977)Google Scholar
  10. 10.
    Pan, J., Fang, K.: Growth Curve Models and Statistical Diagnostics. Springer, New York (2002)zbMATHGoogle Scholar
  11. 11.
    Pan, J., Mackenzie, G.: On modelling mean-covariance structures in longitudinal studies. Biometrika 90(1), 239–244 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Potthoff R. F., Roy S. N.: A generalized multivariate analysis of variance model useful especially for growth curve problems. Biometrika 51(3-4), 313–326 (1964)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Pourahmadi M.: Joint mean–covariance models with applications to longitudinal data: unconstrained parameterisation. Biometrika 86(3), 677–690 (1999)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Xiao, H., Wu, W.: Covariance matrix estimation for stationary time series. Ann. Stat. 40(1), 466–493 (2012)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ye, H., Pan, J.: Modelling of covariance structures in generalised estimating equations for longitudinal data. Biometrika 93(4), 927–941 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Defei Zhang
    • 1
  • Xiangzhao Cui
    • 1
  • Chun Li
    • 1
  • Jianxin Pan
    • 2
    Email author
  1. 1.Department of MathematicsHonghe UniversityMengziChina
  2. 2.Department of MathematicsUniversity of ManchesterManchesterUK

Personalised recommendations