Mobius transformation and Cauchy parameter estimation
成果类型：
Article
署名作者：
McCullagh, P
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
发表日期：
1996
页码：
787-808
关键词：
摘要：
Some properties of the ordinary two-parameter Cauchy family, the circular or wrapped Cauchy family, and their connection via Mobius 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 greater than or equal to 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 less than or equal to 4, and in approximate form for n greater than or equal to 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 Mobius transformation.