Contributions to Nonparametric Predictive
Inference with Right-Censored Data

Abstract

A right-censored data set is most common in reliability and survival analyses.
It occurs when a particular event of interest is not fully observed in an experiment
and when there is no information provided about a random quantity except that it
exceeds a certain value. Nonparametric Predictive Inference (NPI) is a frequentist
statistical method based on only few assumptions. It focuses explicitly on future
observations and uses imprecise probabilities, based on Hill's assumption A(n), to
quantify uncertainty. NPI has been developed for several types of data, including
right-censored data. However, NPI with right-censored data has only taken into
consideration a single future observation.
This thesis presents three contributions to NPI with right-censored data. First,
some statistical methods on extreme values assume that the endpoint of the support
is equal to the largest observed value in a data set. However, a question that may
be of interest is whether, for some right-censored observations in a data set, their
actual value might exceed the largest observed value.
Secondly, the actuarial estimator provides information on the number of events
and censorings at any given discrete point in time. The nature of this estimator
is such that, at every time point (except if all people in the data set have died)
there is right-censoring, the data themselves are not necessarily right-censored. A
similar approach is followed here, but we aim to develop an alternative method to the
actuarial estimator, based on NPI with right-censored data. The proposed method
will be used to derive NPI lower and upper probabilities for a variety of events of interest. As an example application, we apply the newly developed method to obtain
NPI lower and upper survival probabilities for reliability of systems.
Thirdly, NPI has been developed for real-valued data that contain right-censored
observations but only a single future observation was considered. There may be reasons
to be interested in multiple future observations, and it is important that in the
NPI approach such multiple future observations are not conditionally independent
given the data. We extend NPI for right-censored data by considering two future
observations. Particularly, we present NPI lower and upper probabilities for the
event that both future observations are greater than time t. We apply the proposed
method to system reliability.
The results in this thesis widen the applicability of NPI for several real-world
scenarios, while also suggesting new related topics for research.