Let D = {xi} n i=1, where xi ∈ R, be a data set drawn independently from a Gumbel distribution α ∈ R is the location parameter and β > 0 is the scale parameter.

a) Derive an algorithm for estimating α and β.

b) Implement the algorithm derived above and evaluate it on data sets of different sizes. First, find or develop a random number generator that creates a data set with n ∈ {100, 1000, 10000} values using some fixed α and β. Then make at least 10 data sets for each n and estimate the parameters. For each n, report the mean and standard deviation on the estimated α and β. If n = 10000 is too large for your computing resources, skip it.

c) The problem above will require you to implement an iterative estimation procedure. You will need to decide on how to initialize the parameters, how to terminate the estimation process and what the maximum number of iterations should be. Usually, some experimentation will be necessary before you run the experiments in part (b) above. Summarize what you did.