The log-likelihood function for the Cauchy distribution for the sample x1, …, xn is
To find the maximum value of LL we need to solve the following equations simultaneously (the proof requires calculus)
Note too that
We can use Newton’s method to solve these equations, using the estimates from the method of moments as the initial values of µ and σ. Thus, the procedure is
where
We won’t pursue this further here. Instead, we can use the Real Statistics CAUCHY_FIT function that implements the procedure as described in Real Statistics Support for MLE Fitting.
References
Wikipedia (2020) Cauchy distribution
https://en.wikipedia.org/wiki/Cauchy_distribution
AstroML developers (2022) Parameter estimation for the Cauchy (Lorentzen) distribution
https://www.astroml.org/astroML-notebooks/chapter5/astroml_chapter5_Parameter_Estimation_for_Cauchy_Distribution.html