A NOTE ON MIS-SPECIFIED ESTIMATING FUNCTIONS

We consider the use of estimating functions that are not unbiased. Typ- ically, to result in consistent estimators, unbiasedness of estimating functions is a pre-requisite. However, it is sometimes easier to find a useful estimating function that is biased, especially in the presence of missing data or misclassified observa- tions. We show that the root of the estimating function can be modified to give a consistent and asymptotically normal estimator, and illustrate this on several ex- amples with binary data. We compare this to the alternative approach of adjusting the estimating function, and show that it can be more efficient.

[1]  D. Ruppert,et al.  Measurement Error in Nonlinear Models , 1995 .

[2]  V. P. Godambe.  An Optimum Property of Regular Maximum Likelihood Estimation , 1960 .

[3]  D. Cox,et al.  A note on pseudolikelihood constructed from marginal densities , 2004 .

[4]  M. Kenward,et al.  Informative Drop‐Out in Longitudinal Data Analysis , 1994 .

[5]  A. M. Walker.  On the Asymptotic Behaviour of Posterior Distributions , 1969 .

[6]  Wenxin Jiang,et al.  The Indirect Method: Inference Based on Intermediate Statistics—A Synthesis and Examples , 2004 .

[7]  M. Kenward,et al.  Informative dropout in longitudinal data analysis (with discussion) , 1994 .

[8]  Robert Tibshirani,et al.  A Simple Method for the Adjustment of Profile Likelihoods , 1990 .

[9]  R. Jennrich.  Asymptotic Properties of Non-Linear Least Squares Estimators , 1969 .

[10]  J. Robins,et al.  Analysis of semiparametric regression models for repeated outcomes in the presence of missing data , 1995 .

[11]  Bruce W. Turnbull,et al.  Semiparametric Regression Models for Repeated Events with Random Effects and Measurement Error , 1999 .

[12]  A. Rotnitzky,et al.  A note on the bias of estimators with missing data. , 1994, Biometrics.

[13]  Geert Molenberghs,et al.  Regression Models for Longitudinal Binary Responses with Informative Drop‐Outs , 1995 .

[14]  P. J. Huber.  The behavior of maximum likelihood estimates under nonstandard conditions , 1967 .

[15]  H. White.  Maximum Likelihood Estimation of Misspecified Models , 1982 .

[16]  S. Zeger,et al.  Longitudinal data analysis using generalized linear models , 1986 .