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Estimation of the parameters of the extended growth curve model under multivariate skew normal distribution

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  • Jana, Sayantee
  • Balakrishnan, Narayanaswamy
  • Hamid, Jemila S.

Abstract

The Growth Curve Model (GCM) assumes the same shape of profiles for each group, where group means are assumed to be represented by polynomials of the same degree. The model, therefore, is inappropriate when analyzing data from studies involving groups with mean growth curves represented by different shapes. We consider the Extended Growth Curve Model (EGCM), which is a model that allows the group means to follow different degrees of polynomials over time. Existing inference on EGCM assumes multivariate normal errors and produces estimators that are not optimal, when used in the analysis of data with skewed distributions. In this paper, we consider the multivariate skew normal (MSN) distribution as the underlying distribution for the EGCM and provide estimators for its mean and covariance parameters. We adopted the Restricted Expectation–Maximization (REM) algorithm, which is based on the multivariate Newton–Raphson (NR) method and Lagrangian optimization. However, the multivariate NR method and the existing REM algorithm are only applicable to vector parameters and the parameters of interest in this study are matrices. We, therefore, extended the NR approach to matrix parameters, that consequently allowed us to extend the REM algorithm to matrix estimators. The performance of the proposed estimators was examined using extensive simulations and a real data example was also considered to illustrate the application of our proposed estimators.

Suggested Citation

  • Jana, Sayantee & Balakrishnan, Narayanaswamy & Hamid, Jemila S., 2018. "Estimation of the parameters of the extended growth curve model under multivariate skew normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 111-128.
    • Handle: RePEc:eee:jmvana:v:166:y:2018:i:c:p:111-128
    DOI: 10.1016/j.jmva.2018.02.008
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    References listed on IDEAS

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    1. Sayantee Jana & Narayanaswamy Balakrishnan & Dietrich Rosen & Jemila Seid Hamid, 2017. "High dimensional extension of the growth curve model and its application in genetics," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 26(2), pages 273-292, June.
    2. Balakrishnan, N. & Scarpa, Bruno, 2012. "Multivariate measures of skewness for the skew-normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 104(1), pages 73-87, February.
    3. von Rosen, Dietrich, 1989. "Maximum likelihood estimators in multivariate linear normal models," Journal of Multivariate Analysis, Elsevier, vol. 31(2), pages 187-200, November.
    4. Arellano-Valle, Reinaldo B. & Genton, Marc G., 2005. "On fundamental skew distributions," Journal of Multivariate Analysis, Elsevier, vol. 96(1), pages 93-116, September.
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    6. Lin, Tsung I. & Ho, Hsiu J. & Chen, Chiang L., 2009. "Analysis of multivariate skew normal models with incomplete data," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2337-2351, November.
    7. Hamid, Jemila S. & Beyene, Joseph & von Rosen, Dietrich, 2011. "A novel trace test for the mean parameters in a multivariate growth curve model," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 238-251, February.
    8. Jemila Seid Hamid & Dietrich Von Rosen, 2006. "Residuals in the Extended Growth Curve Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(1), pages 121-138, March.
    9. A. Azzalini & A. Capitanio, 1999. "Statistical applications of the multivariate skew normal distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 579-602.
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    1. 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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