The Annals of Statistics

Möbius transformation and Cauchy parameter estimation

Peter McCullagh

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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 $\theta = \mu + i \sigma$ 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 $n \geq 3$, 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 $n \leq 4$, and in approximate form for $n \geq 5$. 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.

Article information

Ann. Statist., Volume 24, Number 2 (1996), 787-808.

First available in Project Euclid: 24 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62A05
Secondary: 62E15: Exact distribution theory

Bartlett adjustment Brownian motion circular Cauchy distribution complex parameter equivariance fractional linear transformation harmonic measure invariant measure invariant statistic likelihood ratio statistic Möbius group robustness wrapped Cauchy distribution


McCullagh, Peter. Möbius transformation and Cauchy parameter estimation. Ann. Statist. 24 (1996), no. 2, 787--808. doi:10.1214/aos/1032894465.

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