I am using GAM models with identity link function and Gaussian family of distributions, more specifically the mgcv package in R. I observed that in this setting, the raw residuals (prediction - real value) are identical to the deviance residuals and the Pearson residuals, and normally distributed. In my understanding, the GAM models assume that the response variable in each observation is drawn from a potentially different distribution within an exponential family, which in my case is Gaussian, and approximate the means of these distributions. I might be confused here, but I don't recall any strong assumptions on the variances of these Gaussians, therefore I don't understand why are the raw residuals equal to the Person or deviance residuals. I imagine the situation in the following way:
The response variable in each observation is drawn from a different Gaussian, with a potentially different variance. I fit a GAM model to the expectations of these Gaussians and compute the raw residuals. For any given observation, the corresponding raw residual tells me how much my prediction is off from the observed value, but that distance has two components. One comes from the error of the model's fit to the expected value for this observation, and the other component comes from the observed value being sampled from this Gaussian distribution instead of simply being equal to the mean. Now, since the variances of the Gaussians for the different observations can be different, it is possible that in some "direction" in the data, this variance is growing. This would lead to a systematic growth in the residuals that is not coming from a lower quality of the fit of the GAM model. If I understand correctly, the normalization in the Person residuals, as well as the deviance residuals, are taking care of such phenomena. In particular, I don't see a reason for any of these residuals to be normally distributed unless stronger assumptions on the shape of the Gaussians is implied. I have been reading Simon Wood's book and I know that some mean-variance relationships are in fact implied, but I feel very confused. Any help with an explanation or a reference would be much appreciated.