Fig 7. Bivariate 3D interpolation of the mean of the Weibull distribution as a function of its two distribution parameters (mean = 0. 6394, var = 0.0417) (left) and negative log likelihood surface of a parametric family of distributions (right).

The negative log-likelihood function has as input arguments the combination of the two parameter values and was used to return the return the negative of this sum. Here the optimization algorithm to which the values are passed searched for minima rather than maxima. Local minimum is printed in red and was estimated using: $(-\log L)=-\log {\displaystyle \prod _{i=1}^{n}f(x_i;a,\beta )=-{\displaystyle \sum _{\iota =1}^n\log f(}}{x}_i;a,\beta )$.