Abstract
In this article, we introduce a conditional marginal model for longitudinal data, in which the residuals form a martingale difference sequence. This model allows us to consider a rich class of estimating equations which contains several estimating equations proposed in the literature. A particular sequence of estimating equations in this class contains a random matrix R *i−1 (β) as a replacement for the “true” conditional correlation matrix of the ith individual. Using the approach of [12], we identify some sufficient conditions under which this particular sequence of equations is asymptotically optimal (in our class). In the second part of the article, we identify a second set of conditions under which we prove the existence and strong consistency of a sequence of estimators of β defined as roots of estimation equations which are martingale transforms (in particular, roots of the sequence of asymptotically optimal equations).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. M. Balan, L. Dumitrescu, and I. Schiopu-Kratina, The Asymptotically Optimal Estimating Equation for longitudinal Data. Strong Consistency, in Tech. Rep. Ser. (Laboratory for Research in Probability and Statistics, Univ. of Ottawa-Carleton Univ., 2008), Vol. 440.
R. M. Balan and I. Schiopu-Kratina, “Asymptotic Results with Generalized Estimating Equations for Longitudinal Data”, Ann. Statist. 33, 522–541 (2005).
G. Casella and R. L. Berger, Statistical Inference, 2nd ed. (Duxbury, 2002).
K. Chen, I. Hu, and Z. Ying, “Strong Consistency of Maximum Quasi-Likelihood Estimators in Generalized Linear Models with Fixed and Adaptive Designs”, Ann. Statist. 27, 1155–1163 (1999).
P. Diggle, P. Heagerty, K.-Y. Liang, and S. Zeger, Analysis of Longitudinal Data, 2nd ed. (Oxford Univ. Press, 2002).
L. Fahrmeir and H. Kaufmann, “Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in Generalized Linear Models”, Ann. Statist. 13, 342–368 (1985).
L. Fahrmeir and H. Kaufmann, “Asymptotic Inference in Discrete Response Models”, Statist. Hefte 27, 179–205 (1986).
J. Fan, T. Huang, and R. Li, “Analysis of Longitudinal Data with Semiparametric Estimation of Covariance Function”, J. Amer. Statist. Assoc. 102, 632–641 (2007).
V. P. Godambe, “An Optimum Property of Regular Maximum Likelihood Estimation”, Ann. Math. Statist. 31, 1208–1211 (1960).
P. Hall and C. C. Heyde, Martingale Limit Theory and Its Applications (Academic Press, 1980).
D. Hedeker and R. D. Gibbons, Longitudinal Data Analysis (Wiley, 2006).
C. C. Heyde, Quasi-Likelihood and its Application. An Optimal Approach to Parameter Estimation (Springer, New York, 1997).
J. Z. Huang, N. Liu, M. Pourahmadi, and L. Liu, “Covariance Selection and Estimation via Penalized Normal Likelihood”, Biometrika 93, 85–98 (2006).
J. Jiang, Y. Luan, and Y.-G. Wang, “Iterative Estimating Equations: Linear Convergence and Asymptotic Properties”, Ann. Statist. 35, 2233–2260 (2007).
H. Kaufmann, “On the Strong Law of Large Numbers for Multivariate Martingales”, Stochastic Processes and Their Applications 26, 73–85 (1987).
T. L. Lai, H. Robbins, and C. Z. Wei, “Strong Consistency of Least Squares Estimates in Multiple Regression. II”, J. Multivar. Anal. 9, 343–361 (1979).
K.-Y. Liang and S. L. Zeger, “Longitudinal Data Analysis Using Generalized Linear Models”, Biometrika 73, 13–22 (1986).
X. Lin and R. J. Carroll, “Nonparametric Function Estimation for Clustered Data when the Predictor is Measured without/with Error”, J. Amer. Statist. Assoc. 95, 520–534 (2000).
J. R. Lockwood and D. F. McCaffrey, “Controlling for Individual Heterogeneity in Longitudinal Models, with Applications to Student Achievement”, Elect. J. Statist. 1, 223–252 (2007).
P. McCullagh and J. A. Nelder, Generalized Linear Models, 2nd ed. (Chapman & Hall, 1989).
J. N. K. Rao, “Marginal Models for Repeated Observatons: Inference with Survey Data”, in Proc. of the Section on Survey Research Methods (Amer. Statist. Assoc., 1998), pp. 76–82.
I. Schiopu-Kratina, “Asymptotic Results for Generalized Estimating Equations with Data from Complex Surveys”, Revue roumaine de math. pures et appl. 48, 327–342 (2003).
J. R. Schott, Matrix Analysis for Statistics, 2nd ed. (Wiley, 2005).
J. Singer and J. Willett, Applied Longitudinal Data Analysis: Modelling Change and Event Occurence (Oxford Univ. Press, 2003).
C. G. Small, J. Wang, and Z. Yang, “Eliminating multiple roots problems in estimation”, Statist. Science 15, 313–341 (2000).
W. K. Thompson, M. Xie, and H. R. Whitem, “Transformations of Covariates for Longitudinal Data”, Biostatistics 4, 353–364 (2003).
N. Wang, R. J. Carroll, and X. Lin, “Efficient Semiparametric Marginal Estimation for Longitudinal/Clustered Data”, J. Amer. Statist. Assoc. 100, 147–157 (2005).
W. B. Wu and M. Pourahmadi, “Nonparametric estimation of large covariance matrices of longitudinal data”, Biometrika 90, 831–844 (2003).
M. Xie and Y. Yang, “Asymptotics for Generalized Estimating Equations with Large Cluster Sizes”, Ann. Statist. 31, 310–347 (2003).
F. Yao, H. G. Müller, and J.-L. Wang, “Functional Data Analysis for Sparse Longitudinal Data”, J. Amer. Statist. Assoc. 100, 577–590 (2005).
F. Yao, H. G. Müller, and J.-L. Wang, “Functional data analysis for longitudinal data”, Ann. Statist. 33, 2873–2903 (2005).
D. Zeng and J. Cai, “Asymptotic Results for Maximum Likelihood Estimators in Joint Analysis of Repeated Measurements and Survival Time”, Ann. Statist. 33, 2132–2163 (2005).
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is based on a portion of the second author’s doctoral thesis.
About this article
Cite this article
Balan, R.M., Dumitrescu, L. & Schiopu-Kratina, I. Asymptotically optimal estimating equation with strongly consistent solutions for longitudinal data. Math. Meth. Stat. 19, 93–120 (2010). https://doi.org/10.3103/S1066530710020018
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066530710020018