A New Approach to Hybrid Censoring
- Ein neuer Zugang zur Hybridzensierung
Górny, Julian; Cramer, Erhard (Thesis advisor); Burkschat, Marco (Thesis advisor); Balakrishnan, Narayanaswamy (Thesis advisor)
Dissertation / PhD Thesis
Dissertation, RWTH Aachen University, 2017
A hybrid censoring model can be described as a censoring procedure whose stopping time is specified by a combination of Type-I censoring and Type-II censoring. The first hybrid censoring model was presented by Epstein (1954) for exponentially distributed lifetimes.This thesis presents basically a new way of approaching hybrid censoring models as well as particular techniques for obtaining the exact distribution of the maximum likelihood estimators (MLEs) for an exponential and for a uniform assumption.In Chapter 1, we present some results for three particular hybrid censoring schemes. Then, the new approach of analyzing hybrid censoring models is described in detail. Following that, we recall in Chapter 2 some basic distributional results for the herein used models of ordered random variables. Chapter 3 gives an introduction into the field of B-splines. After presenting some fundamental properties for divided differences as well as for the B-spline, we derive a new geometric characterization of the univariate B-spline. Further, we derive formulas for particular integrals of B-spline related expressions.Chapters 4 to 8 address several hybrid censoring schemes. First, we consider the Type-I sequential hybrid censoring model (see Chapter 4). There, for different assumptions for the underlying model parameters, basic distributional results as well as the distribution of the MLEs with respect to the two-parameter exponential and the two-parameter uniform distribution are derived. The resulting representations consist in the most cases of B-spline functions. For an exponential assumption an identity is presented, which allows the transition of B-spline based representations to representations consisting of gamma density functions. In Chapter 5, we introduce with the multi-sample Type-I sequential hybrid censoring scheme a hybrid censoring model based on several independently observed samples. Following that, we discuss in Chapters 6 and 7 the Type-II progressive hybrid censoring model and the generalized Type-I and Type-II hybrid censoring schemes, respectively. In Chapter 8, we address the general unified progressive hybrid censoring model. After discussing the respective censoring procedure, we focus in particular on the simplification process of the underlying censoring model.Chapter 9 provides a conclusion as well as an outlook.The appendix consists of elaborations on four additional hybrid censoring models. Further, a brief simulation study, involving all the herein addressed hybrid censoring models, is conducted. The last part of the appendix considers elementary proofs.