I have done a linear regression analysis on some data and I want to construct confidence intervals on the coefficients. I have read that it is necessary for the residuals to be normally distributed to be able to construct accurate confidence intervals for the coefficients. It seems like the residuals of my model are somewhat normally distributed, but would fit better with an Exponentially modified Gaussian distribution (there are a few outliers to the right shifting the mean of the normal distribution). I have a few questions on how to proceed:
- Is it the residuals that need to be normally distributed or is it enough that the errors are assumed to be normally distributed because there is no logical explanation of how the measurement errors couldn't be in the data generation process?
- Can I use the fact that I have fitted a better distribution in any way to create more accurate confidence intervals?
- Does the central limit theorem come into play here? If my sample size is large enough, are the confidence intervals accurate regardless if the residuals are not exactly normally distributed?
- Can I create another model, a generalized linear model, with gaussian family and some link function to make the residuals normally distributed giving better accuracy for the confidence intervals? Or am I misunderstand the use of GLMs.