Nonparametric predictive inference for future order statistics

Abstract

Nonparametric predictive inference (NPI) has been developed for a range of data types, and for a variety of applications and problems in statistics. In this thesis, further theory will be developed on NPI for multiple future observations, with attention to order statistics. The present thesis consists of three main, related contributions. First, new probabilistic theory is presented on NPI for future order statistics; additionally a range of novel statistical inferences using this new theory is discussed. Secondly, NPI for reproducibility is developed by considering two statistical tests based on order statistics. Thirdly, robustness of NPI is introduced which involves a first adaptation of some of the robustness theory concepts within the NPI setting, namely sensitivity curve and breakdown point.

In this thesis, we present NPI for future order statistics. Given data consisting of n real-valued observations, future observations are considered and predictive probabilities are presented for the -th ordered future observation. In addition, joint and conditional probabilities for events involving multiple future order statistics are presented. We further present the use of such predictive probabilities for order statistics in statistical inference, in particular considering pairwise and multiple comparisons based on future order statistics of two or more independent groups of data.



This new theory enables us to develop NPI for the reproducibility of statistical hypothesis tests based on order statistics. Reproducibility of statistical hypothesis tests is an important issue in applied statistics: if the test were repeated, would the same conclusion be reached that is rejection or non-rejection of the null hypothesis? NPI provides a natural framework for such inferences, as its explicitly predictive nature fits well with the core problem formulation of a repeat of the test in the future. For inference on reproducibility of statistical tests, NPI provides lower and upper reproducibility probabilities (RP). The NPI-RP method is presented for two basic tests using order statistics, namely a test for a specific value for a population quantile and a precedence test for comparison of data from two populations, as typically used for experiments involving lifetime data if one wishes to conclude before all observations are available.


As every statistical inference has underlying assumptions about models and specific methods used, one important field in statistics is the study of robustness of inferences. The concept of robust inference is usually aimed at development of inference methods which are not too sensitive to data contamination or to deviations from model assumptions. In this thesis we use it in a slightly narrower sense. For our aims, robustness indicates insensitivity to small changes in the data, as our predictive probabilities for order statistics and statistical inferences involving future observations depend upon the given observations. We introduce some concepts for assessing the robustness of statistical procedures in the NPI framework, namely sensitivity curve and breakdown point. The classical breakdown point does not apply to our context as the predictive inferences are bounded, so we change the definition to suit our context. Most of our nonparametric inferences have a reasonably good robustness with regard to small changes in the data. Traditionally, in the robustness literature there has been quite a lot of emphasis and discussion on robustness properties of estimators for the location parameters. Thus, in our investigation of NPI robustness we also focus on differences in robustness of the mean and the median of the future observations, and see how they relate to the classical concepts of robustness of the median and mean.