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Estimation of parameters under a generalized growth curve model

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  • Filipiak, Katarzyna
  • Klein, Daniel

Abstract

This paper concerns multi-level multivariate data. Such data can be presented in the form of a multi-index matrix (tensor) Y. First the third-order normally distributed tensor of observations, Y∈Rn×p×q, is discussed with the mean structured in the form of a generalized growth curve model, [[X;A,B,C]], with multiplication in all three directions of the third-order tensor X of unknown parameters by the known matrices A, B and C. The paper is focused on the estimation of an unknown tensor X of direct effects and a separable and doubly separable variance–covariance matrix. Since the resulting estimators of unknown parameters cannot be presented in an explicit form, the estimates are obtained approximately. The uniqueness of the so-called ‘flip-flop’ algorithm is also discussed, and the use of the algorithm is illustrated on a real data example. Finally, possible extensions of the third-order generalized growth curve model to more levels are considered.

Suggested Citation

  • Filipiak, Katarzyna & Klein, Daniel, 2017. "Estimation of parameters under a generalized growth curve model," Journal of Multivariate Analysis, Elsevier, vol. 158(C), pages 73-86.
    • Handle: RePEc:eee:jmvana:v:158:y:2017:i:c:p:73-86
    DOI: 10.1016/j.jmva.2017.04.005
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    References listed on IDEAS

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    1. Jushan Bai & Kunpeng Li, 2016. "Maximum Likelihood Estimation and Inference for Approximate Factor Models of High Dimension," The Review of Economics and Statistics, MIT Press, vol. 98(2), pages 298-309, May.
    2. Mitchell, Matthew W. & Genton, Marc G. & Gumpertz, Marcia L., 2006. "A likelihood ratio test for separability of covariances," Journal of Multivariate Analysis, Elsevier, vol. 97(5), pages 1025-1043, May.
    3. Lu, Nelson & Zimmerman, Dale L., 2005. "The likelihood ratio test for a separable covariance matrix," Statistics & Probability Letters, Elsevier, vol. 73(4), pages 449-457, July.
    4. Dayanand Naik & Shantha Rao, 2001. "Analysis of multivariate repeated measures data with a Kronecker product structured covariance matrix," Journal of Applied Statistics, Taylor & Francis Journals, vol. 28(1), pages 91-105.
    5. Markiewicz, Augustyn, 2001. "On dependence structures preserving optimality," Statistics & Probability Letters, Elsevier, vol. 53(4), pages 415-419, July.
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    Cited by:

    1. Shinpei Imori & Dietrich Rosen & Ryoya Oda, 2022. "Growth Curve Model with Bilinear Random Coefficients," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(2), pages 477-508, August.
    2. Pan, Yating & Fei, Yu & Ni, Mingming & Nummi, Tapio & Pan, Jianxin, 2022. "Growth curve mixture models with unknown covariance structures," Journal of Multivariate Analysis, Elsevier, vol. 188(C).

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