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Building adaptive estimating equations when inverse of covariance estimation is difficult

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  • Annie Qu
  • Bruce G. Lindsay

Abstract

Summary. To construct an optimal estimating function by weighting a set of score functions, we must either know or estimate consistently the covariance matrix for the individual scores. In problems with high dimensional correlated data the estimated covariance matrix could be unreliable. The smallest eigenvalues of the covariance matrix will be the most important for weighting the estimating equations, but in high dimensions these will be poorly determined. Generalized estimating equations introduced the idea of a working correlation to minimize such problems. However, it can be difficult to specify the working correlation model correctly. We develop an adaptive estimating equation method which requires no working correlation assumptions. This methodology relies on finding a reliable approximation to the inverse of the variance matrix in the quasi‐likelihood equations. We apply a multivariate generalization of the conjugate gradient method to find estimating equations that preserve the information well at fixed low dimensions. This approach is particularly useful when the estimator of the covariance matrix is singular or close to singular, or impossible to invert owing to its large size.

Suggested Citation

  • Annie Qu & Bruce G. Lindsay, 2003. "Building adaptive estimating equations when inverse of covariance estimation is difficult," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(1), pages 127-142, February.
  • Handle: RePEc:bla:jorssb:v:65:y:2003:i:1:p:127-142
    DOI: 10.1111/1467-9868.00376
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    References listed on IDEAS

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    1. Matyas,Laszlo (ed.), 1999. "Generalized Method of Moments Estimation," Cambridge Books, Cambridge University Press, number 9780521669672, October.
    2. Matyas,Laszlo (ed.), 1999. "Generalized Method of Moments Estimation," Cambridge Books, Cambridge University Press, number 9780521660136, October.
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    Cited by:

    1. Chi-Chuan Yang & Yi-Hau Chen & Hsing-Yi Chang, 2017. "Joint regression analysis of marginal quantile and quantile association: application to longitudinal body mass index in adolescents," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 66(5), pages 1075-1090, November.
    2. Westgate, Philip M., 2013. "A bias-corrected covariance estimator for improved inference when using an unstructured correlation with quadratic inference functions," Statistics & Probability Letters, Elsevier, vol. 83(6), pages 1553-1558.
    3. Ma, Shujie & Liang, Hua & Tsai, Chih-Ling, 2014. "Partially linear single index models for repeated measurements," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 354-375.
    4. Bai, Yang & Fung, Wing K. & Zhu, Zhong Yi, 2009. "Penalized quadratic inference functions for single-index models with longitudinal data," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 152-161, January.
    5. L. Xue & L. Wang & A. Qu, 2010. "Incorporating Correlation for Multivariate Failure Time Data When Cluster Size Is Large," Biometrics, The International Biometric Society, vol. 66(2), pages 393-404, June.
    6. Tze Leung Lai & Dylan Small, 2007. "Marginal regression analysis of longitudinal data with time‐dependent covariates: a generalized method‐of‐moments approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(1), pages 79-99, February.
    7. Annie Qu & Runze Li, 2006. "Quadratic Inference Functions for Varying-Coefficient Models with Longitudinal Data," Biometrics, The International Biometric Society, vol. 62(2), pages 379-391, June.

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