GLM: Sigmoid link with MSE for linear regression? I have a relatively simple regression problem where I wanted to model y given x. X is continuous and is bounded [0,inf); y is bounded (0,1).
My question is, is it appropriate for me to insert a sigmoid link function into my linear regression model?
Predicted values of y should never exceed 1 or go short of 0. The sigmoid function should enforce this.
However, the loss function with be mean squared error.
I think that this is an acceptable GLM, however, I’d like community consensus

It’s no longer a linear model, but it’s fine to use a nonlinear link function with mean squared error loss. Mean squared error loss corresponds to maximum likelihood estimation of a Gaussian likelihood, and the link function does not change that. I have not heard of the nonlinear link function being sigmoid, but I have heard of using a $\log$ link with a Gaussian likelihood (so mean squared error as the loss).
I have my doubts that you really want a Gaussian conditional distribution if you want your $\hat y$ bounded between $0$ and $1$, but that is a separate matter.
