Characterizations of the maximum likelihood estimator of the Cauchy distribution

04/13/2021

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by   Kazuki Okamura, et al.

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We consider characterizations of the maximal likelihood estimator (MLE) of samples from the Cauchy distribution. We characterize the MLE as an attractive fixed point of a holomorphic map on the upper-half plane. We show that the iteration of the holomorphic function starting at every point in the upper-half plane converges to the MLE exponentially fast. We can also characterize the MLE as a unique root in the upper-half plane of a certain univariate polynomial over ℝ. By this polynomial, we can derive the closed-form formulae for samples of size three and four, and furthermore show that for samples of size five, there is no algebraic closed-form formula.