Sampling Distribution of ML Estimators: Cauchy Example

p y 1 s2 + Hx mL2 , where x can have any real value. The distribution has two parameters m and s, which represent its median (the “location” parameter) and semi-interquartile deviation (the “scale” parameter), respectively. This rather unusual distribution has no mean and infinite standard deviation. The exact parameter values are usually not known, and need to be estimated by repeating the corresponding random experiment independently n times, and converting the information thus gathered into two respective estimates of m and s. The best way of doing this is by maximizing the corresponding likelihood function: