Confidence disc and square for Cauchy distributions

TL;DR

This paper develops a method to construct confidence regions for the parameters of Cauchy distributions using complex analysis, providing explicit formulas for disc and square-shaped regions.

Contribution

It introduces a novel approach to simultaneously estimate Cauchy distribution parameters as a complex parameter and derives explicit confidence regions in the complex plane.

Findings

Explicit formulas for confidence disc and square regions.

Unified complex parameter framework for location and scale.

Applicable to samples from Cauchy distributions.

Abstract

We will construct a confidence region of parameters for a sample of size $N$ from Cauchy distributed random variables. Although Cauchy distribution has two parameters, a location parameter $\mu \in \mathbb{R}$ and a scale parameter $\sigma > 0$, we will infer them at once by regarding them as a single complex parameter $\gamma := \mu + i\sigma$. The region should be a domain in the complex plane, and we will give a simple and concrete formula to give the region as a disc and a square.