Characterizations of the maximum likelihood estimator of the Cauchy distribution

Abstract

This paper gives a new approach for the maximum likelihood estimation of the joint of the location and scale of the Cauchy distribution. We regard the joint as a single complex parameter and derive a new form of the likelihood equation of a complex variable. Based on the equation, we provide a new iterative scheme approximating the maximum likelihood estimate. We also handle the equation in an algebraic manner and derive a polynomial containing the maximum likelihood estimate as a root. This algebraic approach provides another scheme approximating the maximum likelihood estimate by root-finding algorithms for polynomials, and furthermore, gives non-existence of closed-form formulae for the case that the sample size is five. We finally provide some numerical examples to show our method is effective.

19 pages; to appear in Lobachevskii Journal of Mathematics

References in corpus (2)

• Bahadur efficiency of the maximum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution
• Limit theorems for quasi-arithmetic means of random variables with applications to point estimations for the Cauchy distribution

Cited by in corpus (4)

• Bahadur efficiency of the maximum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution
• Limit theorems for quasi-arithmetic means of random variables with applications to point estimations for the Cauchy distribution
• Properties of complex-valued power means of random variables and their applications
• Confidence disc and square for Cauchy distributions