The Annals of Statistics

Iterative estimating equations: Linear convergence and asymptotic properties

Jiming Jiang, Yihui Luan, and You-Gan Wang

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We propose an iterative estimating equations procedure for analysis of longitudinal data. We show that, under very mild conditions, the probability that the procedure converges at an exponential rate tends to one as the sample size increases to infinity. Furthermore, we show that the limiting estimator is consistent and asymptotically efficient, as expected. The method applies to semiparametric regression models with unspecified covariances among the observations. In the special case of linear models, the procedure reduces to iterative reweighted least squares. Finite sample performance of the procedure is studied by simulations, and compared with other methods. A numerical example from a medical study is considered to illustrate the application of the method.

Article information

Ann. Statist., Volume 35, Number 5 (2007), 2233-2260.

First available in Project Euclid: 7 November 2007

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Zentralblatt MATH identifier

Primary: 62J02: General nonlinear regression 65B99: None of the above, but in this section 62F12: Asymptotic properties of estimators

Asymptotic efficiency consistency iterative algorithm linear convergence longitudinal data semiparametric regression


Jiang, Jiming; Luan, Yihui; Wang, You-Gan. Iterative estimating equations: Linear convergence and asymptotic properties. Ann. Statist. 35 (2007), no. 5, 2233--2260. doi:10.1214/009053607000000208.

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  • Anderson, T. W. (1971). The Statistical Analysis of Time Series. Wiley, New York.
  • Courant, R. and John, F. (1989). Introduction to Calculus and Analysis. 2. Springer, New York.
  • Diggle, P. J., Liang, K.-Y. and Zeger, S. L. (1994). Analysis of Longitudinal Data. Oxford Univ. Press.
  • Godambe, V. P. (1960). An optimum property of regular maximum likelihood estimation. Ann. Math. Statist. 31 1208–1211.
  • Hand, D. and Crowder, M. (1996). Practical Longitudinal Data Analysis. Chapman and Hall, London.
  • Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models. Chapman and Hall, London.
  • Heyde, C. C. (1997). Quasi-Likelihood and Its Application. Springer, New York.
  • Huber, P. J. (1973). Robust regression: Asymptotics, conjectures and Monte Carlo. Ann. Statist. 1 799–821.
  • Jiang, J. (1996). REML estimation: Asymptotic behavior and related topics. Ann. Statist. 24 255–286.
  • Jiang, J., Luan, Y. and Wang, Y.-G. (2006). Iterative estimating equations: Linear convergence and asymptotic properties. Technical report, Dept. Statistics, Univ. California, Davis.
  • Lehmann, E. L. (1999). Elements of Large-Sample Theory. Springer, New York.
  • Liang, K. Y. and Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika 73 13–22.
  • Luenberger, D. G. (1984). Linear and Nonlinear Programming. Addison-Wesley, Reading, MA.
  • Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992). Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. Cambridge Univ. Press.
  • Richardson, A. M. and Welsh, A. H. (1994). Asymptotic properties of restricted maximum likelihood (REML) estimates for hierarchical mixed linear models. Austral. J. Statist. 36 31–43.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. With Applications to Statistics. Springer, New York.
  • White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica 50 1–25.
  • Wolke, R. and Schwetlick, H. (1988). Iteratively reweighted least squares: Algorithms, convergence analysis and numerical comparisons. SIAM J. Sci. Statist. Comput. 9 907–921.