Agnostic tests can control the type I and type II errors simultaneously

TL;DR

This paper introduces agnostic hypothesis tests that can simultaneously control both type I and type II errors, addressing a key limitation of traditional frequentist tests and improving interpretability.

Contribution

It proposes a framework for constructing agnostic tests that control both errors, including methods to derive such tests from standard p-values and models.

Findings

Agnostic tests can control type I and II errors simultaneously.

Methods to construct unbiased, uniformly most powerful agnostic tests.

Examples of consistent agnostic hypothesis tests.

Abstract

Despite its common practice, statistical hypothesis testing presents challenges in interpretation. For instance, in the standard frequentist framework there is no control of the type II error. As a result, the non-rejection of the null hypothesis cannot reasonably be interpreted as its acceptance. We propose that this dilemma can be overcome by using agnostic hypothesis tests, since they can control the type I and II errors simultaneously. In order to make this idea operational, we show how to obtain agnostic hypothesis in typical models. For instance, we show how to build (unbiased) uniformly most powerful agnostic tests and how to obtain agnostic tests from standard p-values. Also, we present conditions such that the above tests can be made logically coherent. Finally, we present examples of consistent agnostic hypothesis tests.