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 。