Smoothing Functional Data with a Bayesian Hierarchical Model and Robust Fitting of a Weibull Model with Optional Censoring
Cox, Dennis D.
Doctor of Philosophy
In this dissertation, I investigated two independent problems: smoothing functional data with a hierarchical Bayesian model, and robust fitting of a Weibull model for lifetime data with optional right-hand censoring. In the first project, a Bayesian hierarchical model is developed for smoothing functional data. Functional data, with basic data unit being function evaluations (e.g. curves or surfaces) over a continuum, have been frequently encountered in nowadays. While many functional data analysis tools are now available, the issue of simultaneous smoothing is less emphasized. Some methods treat functional data as fully observed while ignoring the measurement noise, others perform smoothing to each functional observation independently thus fail to borrow strength across replications from the same stochastic process. In the first part of this dissertation, a Bayesian hierarchical model is proposed to smooth all functional observations simultaneously. The proposed method relies on priors with data-driven hierarchical parameters, which automatically determine the amount of smoothness. It also provides simultaneous estimates for the mean function and covariance. Case studies of simulated and real data demonstrate that this Bayesian method produces more accurate signal estimates and smooth covariance estimate. In the second project, I explored robust method and estimator for Weibull model. The Weibull family is widely used to model failure data, or lifetime data, although the classical two-parameter Weibull distribution is limited with positive data and monotone failure rate. The parameters of the Weibull model are commonly obtained by maximum likelihood estimation; however, it is well-known that this estimator is not robust when dealing with contaminated data. A new robust way is introduced to fit a Weibull model by using L2 distance, i.e. integrated square distance, of the Weibull probability density function. The Weibull model is augmented with a weight parameter to robustly deal with contaminated data. Results comparing a maximum likelihood estimator with an L2 estimator are given in this article, based on both simulated and real data sets. It is shown that this new L2 parametric estimation method is more robust and does a better job than maximum likelihood in the newly proposed Weibull model when data are contaminated. The same preference for L2 distance criterion and the new Weibull model also happens for right censored data with contamination.
Smoothing; Functional data; Bayesian hierarchical model; Robust; Weibull model