I am interested in fitting the following Gaussian process regression model: y_i = f(x_i) + e_i, where x_i is a p-dimensional vector of covariates, and e_i is iid N(0, s2). Note that f(.) is an unkown function of the covariates, and I want to use gaussian process regression to nonparametrically estimate f, then test if f is significantly related to y(i.e. is the mean of f different than 0? is its variance different than 0?). Basically I want to use GP regression to conduct a nonparametric global omnibus test to see if my covariates X are significantly related to Y.
I notice in the gaussian process regression literature that the noise is often assumed to be iid N(0, s2), however, this assumption may be unrealistic in practice. It is my understanding that if this noise assumption is violated, then any testing I do of f will likely have poor frequentest coverage rates (i.e. inflated type-1-error rates).
Therefore, my question is as follows: is anyone aware of any literature that discusses the properties of the gaussian process regression model given model misspecification (e.g. when the noise assumption is incorrect)? In particular, is it possible to still conduct valid inferences on f even when my noise assumption is violated? If not, is there anyway to modify the GP model (perhaps through some sort of robust sandwich covariance estimator) to obtain more robust inferences on f given non iid noise?