Confidence disc and square for 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.

v4, 15 pages, 6 figures, typo in Eq. (3) corrected, some unnecessary columns in Figures 2,3,5 and 6 deleted, to appear in Ukrainian Mathematical Journal

References in corpus (3)

• Bahadur efficiency of the maximum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution
• Characterizations of the maximum likelihood estimator of the Cauchy distribution
• Limit theorems for quasi-arithmetic means of random variables with applications to point estimations for the Cauchy distribution

Cited by in corpus (1)

• Properties of complex-valued power means of random variables and their applications