In this paper, the author try to fit the Gumbel distribution based on the r largest value of each year using the maximal likelihood estimators: the likelihood function for r largest values $X_{n1},\dots, X_{nr}$ of each year $n=1,\dots, N$.
Since there is a increasing trend of the data (not iid), so we cannot use the likelihood function for iid data.
Question: How do we deal this case?
Instead of assuming a fixed $\mu$, the authors use the regression function $\mu(n) = \alpha +\beta n/N$. So they plug its RHS into the ikelihood, to replace each $\mu$, and get a model with three unknown parameters $\sigma, \alpha, \beta$. You estimate them the usual way, by maximising the likelihood (2.5) for your data at the design points. You can use a standard optimisation routine, such as optim in R. (Note the need for constraints, such as $\sigma \geq tol > 0$.) Typically, handling a $\log$ likelihood is easier numerically. There is R functionality for the standard Gumbel distribution, which could help with a starting guess in optimisation. There might be a more specific R package for this type of problem.
