Emax Type I Extreme Value / Generalized Extreme Value

emax.m

% Mean utilities
mu1 = 2;
mu2 = 3;
mu3 = 1;

% Scale parameter
sigma = 0.4;

% Draw Type I extreme value errors 
R = -evrnd(0,1,100000,3);

% Add mean utilities to errors
R(:,1) = sigma*R(:,1)+mu1;
R(:,2) = sigma*R(:,2)+mu2;
R(:,3) = sigma*R(:,3)+mu3;

% Compute the unconditional means
mean(R,1)
[mu1, mu2, mu3] + 0.577*sigma

% Compute the unconditional st. deviations
std(R,1)
sqrt(sigma^2*pi^2/6)

% Compute the Emax numerically
mean(max(R,[],2))

% Compute the Emax analytically
sigma*0.577 + sigma*log(exp(mu1/sigma) + exp(mu2/sigma) + exp(mu3/sigma))

% Let's make this numerically safe
mu1 + sigma*0.577 + sigma*log(1 + exp((mu2-mu1)/sigma) + exp((mu3-mu1)/sigma))