Parametric Predictive Bootstrap and Test Reproducibility

Abstract

Bootstrap methods have become one of the most widely used statistical techniques due to their simplicity and good properties. In this thesis, we introduce a novel bootstrap method which we call the parametric predictive bootstrap (PP-B). The PP-B method relies on parametric models, and it is primarily designed for predictive inference. In the PP-B method, a single observation is sampled from the assumed distribution with estimated parameters based on an available data set of size n. Then, this observation is added to the data and the process is repeated, now with n+1 observations. This process continues to sample in total m values in the same way, each observation being added to the data and re-estimating the parameters before sampling the next observation. The PP-B sample consists of m newly drawn observations and excludes the n original data observations. The performance of the PP-B method is studied on finite and infinite data ranges, and compared to other bootstrap methods via simulations, which show that it works well as a method for predictive inference. The PP-B method is applied to a range of scenarios to evaluate its performance. It relies on an assumed parametric model and we examine how it performs when the model is misspecified.

A hypothesis test is one of the most important tools in the practical application of statistics. Statistical hypothesis tests can have different results when they are repeated. The reproducibility probability of hypothesis tests has gained increasing attention due to its importance in evaluating the variability and the stability of test results. The PP-B method is presented for the reproducibility probability (RP) of some parametric tests. Test reproducibility is naturally regarded as a predictive inference problem, which is consistent with the PP-B method. The explicitly predictive nature of PP-B provides an appropriate formulation for inferring RP, as the nature of RP is explicitly predictive as well. The performance of PP-B for RP is compared with the nonparametric predictive inference bootstrap method, which also has a predictive nature but does not assume a parametric model.