Date of Award

Spring 5-31-2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Antai Wang

Second Advisor

Sunil Kumar Dhar

Third Advisor

Ji Meng Loh

Fourth Advisor

Sundarraman Subramanian

Fifth Advisor

Zhi Wei

Abstract

This dissertation has three independent parts. The first part studies a variation of the competing risks problem, known as the semi-competing risks problem, in which a terminal event censors a non-terminal event, but not vice versa, in the presence of a censoring event which is independent of these two events. The joint distribution of the two dependent events is formulated under Archimedean copula. An estimator for the association parameter of the copula is proposed, which is shown to be consistent. Simulation shows that the method works well with most common Archimedean copula models.

The second part studies the properties of a special class of frailty models when the frailty is common to several failure times. The model is closely linked to Archimedean copula models. A useful formula for baseline hazard functions for this class of frailty models is established. A new estimator for baseline hazard functions in bivariate frailty models based on dependent censored data with covariates is obtained, and a model checking procedure is presented.

The third part studies the properties of frailty models for bivariate data under fixed left censoring. It turns out that the distribution of observable pairs belongs to a new class of bivariate frailty models. Both the original model for complete data and the new model for observable pairs are members of Archimedean copula family. A new estimation strategy to analyze left-censored data using the corresponding Kendalls distribution is established.

Included in

Mathematics Commons

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