Möbius transformation and Cauchy parameter estimation

Abstract

Some properties of the ordinary two-parameter Cauchy family, the circular or wrapped Cauchy family, and their connection via Möbius transformation are discussed. A key simplification is achieved by taking the parameter θ = μ + iσ to be a point in the complex plane rather than the real plane. Maximum likelihood estimation is studied in some detail. It is shown that the density of any equivariant estimator is harmonic on the upper half-plane. In consequence, the maximum likelihood estimator is unbiased for n ≥ 3, and every harmonic or analytic function of the maximum likelihood estimator is unbiased if its expectation is finite. The joint density of the maximum likelihood estimator is obtained in exact closed form for samples of size n ≤ 4, and in approximate form for n ≥ 5. Various marginal distributions, including that of Student's pivotal ratio, are also obtained. Most results obtained in the context of the real Cauchy family also apply to the wrapped Cauchy family by Möbius transformation.