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 residuals in gamlss are normalised quantile residuals
These residuals have a standard normal distribution for the true model, so compare these residuals with a standard normal distribution.
The easiest way in gamlss is
plot(m1)

which gives you a histogram and a normal QQ plot of these residuals.
You can also get a de-trended normal QQ plot of the residuals in gamlss (called a worm plot) by
wp(m1)

I don't think these residuals are available in gam. I like the gamlss residuals because they are easy to check against a standard normal distribution.
Other residuals like: "deviance", "pearson","scaled.pearson", "working", or "response" are not normally distributed, and so are difficult to check.
