Adaptive and Multistage Procedures for Inference on Monotone RegressionFunctions in Designd Studies and Observed Data Settings

[摘要] This dissertation consists of three works. The first work is on the estimation and hypothesis testing for a failure time distribution function at a point in the current status model with observation times supported on a grid of unknown sparsity and with multiple subjects sharing an observation time. The grid resolution is specified as $c n^{-gamma}$ with $c, gamma > 0$.The asymptotic behavior falls into three cases depending on $gamma$: regular Gaussian-type asymptotics obtain for $gamma < 1/3$, non-standard cube-root Chernoff-type asymptotics prevail when $gamma > 1/3$ and $gamma=1/3$ serves as a boundary case. The boundary limit distribution indexed by $c$ is different from the Gaussian and Chernoff distributions in the previous cases but converges weakly to them as $c$ goes to $infty$ and $0$, respectively. This relationship among the limit distributions allows us to develop an adaptive procedure to construct confidence intervals for the value of the failure time distribution at a point without estimating $gamma$, which is of enormous advantage from the perspective of inference. In the second work, we consider a hybrid two-stage procedure (TSP) for estimating an inverse regression function at a given point, where isotonic regression is first implemented to obtain an initial estimate and then a local linear approximation is exploited over the vicinity of this estimate. The convergence rate of the second-stage estimate can attain the parametric rate $n^{1/2}$. Furthermore, a bootstrapped variant of TSP is introduced and its consistency properties established. This variant manages to overcome the slow speed of the convergence in distribution and the estimation of unknown parameters.The third work shares the same basic problem with the second work. In practice, the previous hybrid two-stage procedure usually results in a biased estimator if the sample size is small and the regression function is very locally nonlinear around the target point. For such cases, we propose to adopt isotonic regression and smoothed isotonic regression at stage two and denote the resulting two-stage procedures by TSIRP and TSSIRP, respectively. The convergence rate of the estimate from TSIRP is less than the parametric rate while that from TSSIRP can achieve it.

[发布机构] University of Michigan

[效力级别] Current Status Model

[关键词] Isotonic Regression;Current Status Model;Time Sparsity;Local Linear Approximation;Bootstrap;Two-stage Procedure;Statistics and Numeric Data;Science;Statistics