Penalized unimodal spline density estimate with application to M-estimation - Chen, Xin, author; Meyer, Mary C., advisor; Wang, Haonan, committee member; Kokoszka, Piotr, committee member; Zhou, Wen, committee member; Miao, Hong, committee member (Colorado State University. Libraries, 2020)
This dissertation establishes a novel type of robust estimation, Auto-Adaptive M-estimation (AAME), based on a new density estimation. The new robust estimation, AAME, is highly data-driven, without the need of priori of the error distribution. It presents improved performance against fat-tailed or highly-contaminated errors over existing M-estimators, by down-weighting influential outliers automatically. It is shown to be root-n consistent, and has an asymptotically normal sampling distribution which provides asymptotic confidence intervals and the basis of robust prediction intervals. The new density estimation is a penalized unimodal spline density estimation which is established as a basis for AAME. It is constrained to be unimodal, symmetrical, and integrate to 1, and it is penalized to have stabilized derivatives and against over-fitting, overall satisfying the requirements of being applied in AAME. The new density estimation is shown to be consistent, and its optimal asymptotic convergence rate can be obtained when the penalty is asymptotically bounded. We also extend our AAME to linear models with heavy-tailed and dependent errors. The dependency of errors is modeled by an autoregressive process, and parameters are estimated jointly.
Keywords: splines; robust estimation; unimodal density