GAM with non-normal noise I am starting to work with GAM models, and I am trying to figure out the consequences of having non-normal noise. Consider the following example (just note that Gumbel(-0.57, 1) is a centered continued distribution with a finite variance, but skewed):
n=1000;
X1=rexp(n); X2=runif(n, 0, 5); Y=-2 + sin(X1) + X2^3 + rgumbel(n, -0.57, 1);

fit=gam(Y~s(X1)+s(X2))

So, can I somehow take into account that the noise is non-normal? Do I have to?
Some motivation: I am actually trying to estimate the distribution of the noise variables here (as they have a specific interpretation in my problem). So I would like to see something like fit$residuals having approximately Gumbel distribution. Is it true though that as $n\to\infty$ we would not reject ks.test(fit$residuals, Gumbel)? Or in other words, are the estimations of $\hat{f}_1,\hat{f}_2$ consistent even if we don't have normal noise?

The following command in the gamlss R package
m1 <- gamlss(Y ~ pb(X1) + pb(X2), family = GU)

fits Y ~ GU(mu, sigma) distribution where mu = pb(X1) + pb(X2),  i.e. a smooth function of X1 + a smooth function of X2. The amount of smoothing (or degrees of freedom used for smoothing) is calculated automatically, and log(sigma) = constant, because log is the default link function for sigma for GU, so sigma is also constant, and the constant is fitted.
If you want to fix sigma at a specific value, e.g., 2, use
m1 <- gamlss(Y ~ pb(X1) + pb(X2), sigma.start=2, sigma.fix=TRUE, family = GU)

If you want to fit a model for log(sigma), use e.g.,
m1 <- gamlss(Y ~ pb(X1) + pb(X2), sigma.fo =~ pb(X1) + pb(X2), family = GU)

(Note sigma.fo is short for sigma.formula.)
