The Annals of Statistics

Möbius transformation and Cauchy parameter estimation

Peter McCullagh

Full-text: Open access

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 $\theta = \mu + i \sigma$ 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

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

Dates
First available in Project Euclid: 24 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1032894465

Digital Object Identifier
doi:10.1214/aos/1032894465

Mathematical Reviews number (MathSciNet)
MR1394988

Zentralblatt MATH identifier
0859.62007

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

Keywords
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

Citation

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


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References

  • COPAS, J. 1975. On the unimodality of the likelihood function for the Cauchy distribution. Biometrika 62 701 704. Z.
  • EATON, M. L. 1989. Group Invariance Applications in Statistics. IMS, Hay ward, CA. Z.
  • FERGUSON, T. 1978. Maximum likelihood estimates of the parameters of the Cauchy distribution for samples of size 3 and 4. J. Amer. Statist. Assoc. 73 211 213. Z.
  • KNIGHT, F. B. 1976. A characterization of the Cauchy ty pe. Proc. Amer. Math. Soc. 55 130 135. Z.
  • KNIGHT, F. B. and MEy ER, P. A. 1976. Characterisation de la loi de Cauchy. Z. Wahrsch. Verw. ´ Gebiete 34 129 134. Z.
  • KOLASSA, J. and MCCULLAGH, P. 1990. Edgeworth series for lattice distributions. Ann. Statist. 18 981 985. Z.
  • MARDIA, K. V. 1972. Statistics of Directional Data. Academic Press, New York.
  • MCCULLAGH, P. 1992. Conditional inference and Cauchy models. Biometrika 79 247 259. Z.
  • MCCULLAGH, P. 1993. On the choice of ancillary in the Cauchy location-scale problem. In Z Statistics and Probability: A R. R. Bahadur Festschrift J. K. Ghosh, S. K. Mitra, K. R.. Partasarathy and B. L. S. Prakasa Rao, eds. 445 462. Wiley Eastern, New Delhi. Z.
  • ROGERS, L. C. G. and WILLIAMS, D. 1986. Diffusions, Markov Processes, and Martingales 2. Wiley, New York. Z.
  • RUDIN, W. 1987. Real and Complex Analy sis. McGraw-Hill, New York. Z.
  • WOLD, H. 1934. Sheppard's correction formulae in several variables. Skand. Akturarietidskrift 17 248 255.
  • CHICAGO, ILLINOIS 60637