In many statistical applications the number of nuisance parameters increases with the number of observations.  For example, in a matched study, the nuisance parameters are stratum-specifid; alternatively, in a study with errors-in-covariates, the unknown true values of mismeasured covariates can be considered as subject-specific nuisance parameters.  The presence of many nuisance parameters can cause inconsistency of the estimators of parameters of interest, which is well knows as the Neyman & Scott (1948) problem.  Our research focuses on methods to reduce the impact of nuisance parameters on estimating the parameters of interest in a general estimation function framework.

At the heart of the problem an estimating function for the interest parameters, which is unbiased when the nuisance parameters are known, might become severely biased in the more realistic situation where one substitutes and estimator of  ("profiles out") the unknown nuisance parameters in the estimating function.  

We propose a simple adjustment to the profile estimating function, derived from a Taylor series expansion, that achieves first order bias correction of the profile estimating function.  This adjustment method can be used in both parametric and semiparametric settings.  In the semiparametric setting, typically only the first two moments of the response variable are needed to form the adjustment. Theoretical properties of the proposed adjusted profile estimating function include a close connection tot he adjusted profile score approach of Cox & Reid (1987) and invariance under reparameterizations of the parameters of interest and nuisance parameters.  Important applications of this method include the estimation of the pairwise association in stratified, clustered data and estimation of the main effects in a matched pair study.  A Brief simulation study shows that the proposed method considerably reduces the impact of the nuisance parameters.  In another application of this method, we present a way to jointly estimate the intrafamilial resemblance and main effects for matched case-control family studies with adjustment for stratum-specific nuisance effects and non-random ascertainment of families.

The proposed adjusted profile estimating function method is typically not applicable to a study with errors-in-covaiates when information about variance of the surrogate is incomplete and cannot be modeled accurately.  We provide an orthogonality condition under which a second-order locally ancillary estimating function can achieve first-order bias correction without requiring full knowledge of the variance of the data.  Adopting this orthogonality condition, we propose second-order locally ancillary estimating function for studies with errors-in-covariates where the mismeasured unknown covariates are treated as fixed nuisance parameters.  The proposed estimating function is applicable with the information about the second moment of the data is incomplete and the second moment cannot be modeled accurately.  This approach can also be used in matched pair studies where the dispersion parameter is unknown.  Theory and simulation studies show that the proposed estimating function substantially reduces the impact of the nuisance parameters.  We apply the method to a coronary heart disease study with a mismeasured blood pressure covariate.

The proposed adjusted profile estimating function method and orthogonal second-order locally ancillary estimating function method are compared both analytically and using simulations.