Elucidating Inferential Models with the Cauchy Distribution

TL;DR

This paper explores various statistical inference methods using the Cauchy distribution as a challenging example, demonstrating that Inferential Models provide exact, efficient, and prior-free inference for its parameters.

Contribution

It shows that Inferential Models outperform other methods in providing exact and efficient inference for the Cauchy distribution's parameters.

Findings

Inferential Models yield exact inference for Cauchy parameters.

Other methods face difficulties in inference accuracy.

Inferential Models are prior-free and efficient.

Abstract

Statistical inference as a formal scientific method to covert experience to knowledge has proven to be elusively difficult. While frequentist and Bayesian methodologies have been accepted in the contemporary era as two dominant schools of thought, it has been a good part of the last hundred years to see growing interests in development of more sound methods, both philosophically, in terms of scientific meaning of inference, and mathematically, in terms of exactness and efficiency. These include Fisher's fiducial argument, the Dempster-Shafe theory of belief functions, generalized fiducial, Confidence Distributions, and the most recently proposed inferential framework, called Inferential Models. Since it is notoriously challenging to make exact and efficient inference about the Cauchy distribution, this article takes it as an example to elucidate different schools of thought on statistical inference. It is shown that the standard approach of Inferential Models produces exact and efficient prior-free probabilistic inference on the location and scale parameters of the Cauchy distribution, whereas all other existing methods suffer from various difficulties.