Three Improvements of Confidence Intervals

The distinction between descriptive statistics and statistical modelling via probability theory is not always as useful as presented in textbooks. In this paper we take confidence intervals as an example. These intervals are defined for statistical models and are used for hypothesis testing. We propose to define slightly different intervals based on ideas based on cross-validation and on the bootstrap method. These intervals will be defined in a purely descriptive manner and that has an interpretation directly related to the sample at hand. Although these new resampling intervals have descriptive definitions they can be used for testing in just the same manner as confidence intervals. The new intervals are of particular relevance in cases where the sample cannot be modeled by an infinite sequence of iid variables. In order to compare cover probabilities we suggest to randomize the sample size. We demonstrate that both the new resampling intervals as well as z-intervals, Wilson intervals, and likelihood ratio intervals get better coverage probabilities if we use additive smoothing.