Nonparametric Predictive Methods for Bootstrap and Test Reproducibility

Abstract

This thesis investigates a new bootstrap method, this method is called Nonparametric
Predictive Inference Bootstrap (NPI-B). Nonparametric predictive inference
(NPI) is a frequentist statistics approach that makes few assumptions, enabled by
using lower and upper probabilities to quantify uncertainty, and explicitly focuses
on future observations. In the NPI-B method, we use a sample of n observations to
create n + 1 intervals and draw one future value uniformly from one interval. Then
this value is added to the data and the process is repeated, now with n+1 observations.
Repetition of this process leads to the NPI-B sample, which therefore is not
taken from the actual sample, but consists of values in the whole range of possible
observations, also going beyond the range of the actual sample. We explore NPI-B
for data on finite intervals, real line and non negative observations, and compare
it to other bootstrap methods via simulation studies which show that the NPI-B
method works well as a prediction method.
The NPI method is presented for the reproducibility probability (RP) of some
nonparametric tests. Recently, there has been substantial interest in the reproducibility
probability, where not only its estimation but also its actual definition
and interpretation are not uniquely determined in the classical frequentist statistics
framework. The explicitly predictive nature of NPI provides a natural formulation
of inferences on RP. It is used to derive lower and upper bounds of RP values (known
as the NPI-RP method) but if we consider large sample sizes, the computation of
these bounds is difficult. We explore the NPI-B method to predict the RP values
(they are called NPI-B-RP values) of some nonparametric tests. Reproducibility of
tests is an important characteristic of the practical relevance of test outcomes.