On $p$-values

TL;DR

This paper discusses the interpretation of p-values as measures of model approximation quality, emphasizing that models are approximations rather than true representations, and introduces approximation regions distinct from confidence regions.

Contribution

It introduces the concept of approximation regions for model assessment and clarifies the interpretation of p-values as indicators of approximation quality.

Findings

Small p-values indicate poor model approximation

Approximation regions are distinct from confidence regions

Models are treated as approximations, not true representations

Abstract

Models are consistently treated as approximations and all procedures are consistent with this. They do not treat the model as being true. In this context $p$-values are one measure of approximation, a small $p$-value indicating a poor approximation. Approximation regions are defined and distinguished from confidence regions.