Estimation Theory for Semiparametric Models with Bundled Parameters

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The investigator proposes extensions of existing asymptotic distributional theories for M- and Z-estimations in the semiparametric models with separated parameters to accommodate situations where the estimation criteria for the semiparametric models are parameterized with bundled parameters, i.e. the infinite dimensional parameter is an unknown function of the parameter of interest. The proposal is motivated by several statistical problems including the efficient estimation in the linear regression model with censored data under several different censoring mechanisms, the efficient estimation in the single index model, the partial likelihood estimation in the Cox regression model with an unknown link function, and the weighted estimation for missing data problems in survival analysis. The investigator also proposes to apply the general theory for bundled parameters to all these problems, particularly for the case that the infinite dimensional nuisance parameter is approximated by regression splines.The proposed research is primarily motivated by PI's collaboration in biomedical studies, where more robust statistical modeling techniques are desirable to reduce the uncertainty of model misspecification, particularly when data are incomplete due to limited study follow-up. The proposed research will also allow the investigator to add more thorough statistical results to the course of advanced survival analysis and be helpful in developing the special topic course on semiparametric models into a regular Ph.D. level course. The proposed research activities will motivate graduate students to become independent researchers who are able to engage in fundamental statistical research.