Mathematical Sciences: Bandwidth Selection in Semiparametric Regression Problems

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9423247 Wang Abstract This project investigates bootstrap bandwidth selection procedures in semiparametric regression models. As pointed out in Hsieh and Manski (1987), the performance of the estimated regression parameter in a semiparametric regression model could be rather sensitive to the choice of bandwidth. The traditional plug-in method often requires estimating elements within a complicated second order expansion of the estimated regression parameter. The fact that the properly chosen bootstrap procedure could estimate these complicated terms in an automatic fashion makes this procedure especially appealing. Two classes of semiparametric regression models are considered here: (i) semiparametric heteroscedastic regression models, and (ii) semiparametric measurement error regression models. Different resampling strategies from those in the parametric and nonparametric regressions have to be pursued due to the different semiparametric structure. Resampling strategies for the aforementioned two semiparametric settings are proposed. Properties of the proposed bandwidth estimators are investigated theoretically as well as numerically. The availability of high-speed computing has allowed researchers to use "semiparametric modeling," a new mathematical development which, for most scientific explorations, is preferable to its traditional parametric rivals because of much greater flexibility. Some recent applications of semiparametric modeling include studies which compare the effects of high versus low AZT treatments on the survival of AIDS patients, the investigation of effects of saturated fat on breast cancer based on self-support food questionnaires and on detailed food records, the relationship between a patient's age at the time of bone-marrow transplant and the incidence of chronic graft versus host disease (GVHD) afterwards. One of the major difficulties in semiparametric modeling is to properly choose a so-called "bandwidth" which controls the amount of data going into the model simultaneously.. Poor choice can result in loss of information or loss of accuracy. This project investigates bandwidth selection procedures which would be theoretically sound and practically feasible. The main approach emphasizes taking advantage of modern computing power to help users to avoid complicated mathematical derivations. Success of this project will certainly further promote the applications of semiparametric modeling in many other fields.