Title: Applications of influence functions to semiparametric regression models
Description: Linear and loglinear semiparametric regression models are commonly used for the effect of particulate matter suspended in the air on mortality [Dominici et al., 2004]. Commonly used estimation methods include series linear and Poisson regression estimators. The use of such "first order" estimators leads to complications in analysis when certain nuisance parameters do not have a sufficient number of derivatives, including squared bias that is larger than the variance and consequently, invalid confidence intervals. "Second order" estimators [Robins et al., 2008] based on second order influence functions perform better than the first order estimators when the nuisance parameters are rough. We describe and prove bias and variance properties of these estimators for i.i.d. data. Time series data introduces additional complications for these estimators because of correlations between observations. We modify the second order influence function estimators to account for time. We identify differences between the two situations and establish conditions under which analytical results on bias and variance can be obtained for the time series estimators. We study the effects of various tuning parameters on the performance of the estimators and provide a test for large bias of the first order estimator. We apply the methods developed here to National Morbidity, Mortality and Air Pollution Study (NMMAPS) data on Minneapolis and find evidence for bias in the first order estimates.

We next turn to the problem of semiparametric quantile regression for censored survival data. Existing methods for censored quantile regression ( [Portnoy, 2003] and [Peng and Huang, 2008]) are consistent and asymptotically normal under the relatively stringent assumption of a linear or loglinear model for all quantiles of the survival distribution. We impose a modeling assumption only on the quantile of interest, and use the method of targeted maximum likelihood to estimate the regression coefficients in a counterfactual data setting. We show that the estimator has double-robustness properties and compare the performance of the estimator with competing estimators through simulations.