Mathematical Sciences: Bandwidth Selection in Semiparametric Regression Problems

9423247王摘要本项目研究半参数回归模型中的Bootstrap带宽选择过程。正如Hsieh和Manski(1987)所指出的，在半参数回归模型中，估计回归参数的性能可能对带宽的选择相当敏感。传统的插值法通常需要估计回归参数的复杂二阶展开式内的元素。正确选择的引导过程可以自动估计这些复杂的术语，这一事实使该过程特别有吸引力。本文考虑两类半参数回归模型：(I)半参数异方差回归模型和(Ii)半参数测量误差回归模型。由于半参数结构的不同，必须采用与参数回归和非参数回归不同的重采样策略。提出了上述两种半参数设置的重采样策略。对所提出的带宽估计器的性质进行了理论和数值研究。高速计算的可获得性使研究人员能够使用“半参数建模”，这是一种新的数学发展，对于大多数科学探索来说，由于其更大的灵活性，它比传统的参数竞争对手更可取。半参数模型的一些最新应用包括比较高和低AZT治疗对艾滋病患者存活率的影响的研究，基于自给式食物问卷和详细食物记录的饱和脂肪对乳腺癌的影响的调查，以及患者在骨髓移植时的年龄与之后慢性移植物抗宿主病(GVHD)发病率的关系。半参数建模的主要困难之一是正确选择一个所谓的“带宽”，它控制同时进入模型的数据量。错误的选择可能会导致信息丢失或准确性下降。该项目研究的是理论上合理且实际可行的带宽选择程序。主要方法强调利用现代计算能力来帮助用户避免复杂的数学推导。该项目的成功必将进一步推动半参数建模在其他许多领域的应用。

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.