Three-step N–R algorithm for the maximum-likelihood estimation of the general extreme values distribution parameters

First, starting out with the magnitudes of both the scale and location parameters given by the method of moments (MOMs), and assuming they are constants, the root of the single equation formed by equating the partial derivative of the log-likelihood function (LLF) with respect to the shape parameter (a) to zero is solved, ∂LLF/∂a = 0. Next, with this value for the shape parameter and the MOMs estimate for the location parameter (c), the root of the single equation formed by equating the partial derivative of the LLF with respect to the scale parameter (b) to zero is solved, ∂LLF/∂b = 0. Next, with the recently computed values for the shape and scale parameters, the root of the single equation formed by equating the partial derivative of the LLF with respect to the location parameter to zero is solved, ∂LLF/∂c = 0. Next, with the recently computed values for the other two parameters, these single equations are solved once again, in the same order. As a result of these two cycles, the roots of the single equations turn out to be close to the roots of the simultaneous system of equations of ∂LLF/∂a = 0 and ∂LLF/∂b = 0 and ∂LLF/∂c = 0. Finally, the system of three equations is solved by the Newton–Raphson (N–R) algorithm with those values used as initial estimates.