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.

翻译：本文给出了对Cauchy分布分布位置和比例的组合最大可能性估计的新方法。 我们将联合视为一个单一的复杂参数, 并得出一个复杂变量的可能性方程式的新形式。 基于此方程式, 我们提供一个新的迭代方案, 与最大可能性估计相近。 我们还以代数方式处理该方程式, 并得出一个包含最大可能性估计值作为根数的多元分子。 这种代数法提供了另一个方案, 接近于对多元数值的根值算法的最大可能性估计值, 此外, 我们为样本大小为五的情况提供了封闭式公式的不存在。 我们最后提供了一些数字示例, 以显示我们的方法是有效的 。