10 - General Methods for Frequentist Inferences

Summary

This chapter briefly covers many general methods for calculating p-values and confidence intervals. We discuss likelihood ratios for inferences with a one-dimensional parameter. Pivot functions are defined (e.g., the probability integral transformation). Basic results for normal and asymptotic normal inferences are given, such as some central limit theorems and the delta method. Three important likelihood-based asymptotic methods (the Wald, score, and likelihood ratio test) are defined and compared. We describe the sandwich method for estimating variance, which requires fewer assumptions than the likelihood-based methods. General permutation tests are presented, along with implementation descriptions including equivalent forms, permutational central limit theorem, and Monte Carlo methods. The nonparametric bootstrap is described, as well as some bootstrap confidence interval methods such as the BCa methods. The melding method of combining two confidence intervals is described, which gives an automatically efficient method to calculate confidence intervals for the differences or ratios of two parameters. Finally, we discuss within-cluster resampling.