Fitting Cauchy Parameters via MLE

The log-likelihood function for the Cauchy distribution for the sample x1, …, xn is

LL Cauchy distribution

To find the maximum value of LL we need to solve the following equations simultaneously (the proof requires calculus)

Simultaneous equations

Note too that

Bounds for sigma

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

Initial guess

Step i+1

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

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