Solving just for x_0 requires solving a polynomial of degree 2n-1, and solving just for \,\!\gamma requires solving a polynomial of degree 2n. Therefore, whether solving for one parameter or for both parameters simultaneously, a numerical solution on a computer is typically required. The benefit of maximum likelihood estimation is asymptotic efficiency; estimating x_0 using the sample median is only about 81% as asymptotically efficient as estimating x_0 by maximum likelihood. The truncated sample mean using the middle 24% order statistics is about 88% as asymptotically efficient an estimator of x_0 as the maximum likelihood estimate. When Newton's method is used to find the solution for the maximum likelihood estimate, the middle 24% order statistics can be used as an initial solution for x_0.
