Uniformly Distributed Residuals in Linear Regression What can you say about your linear regression if the residuals are uniformly distibuted (and not normal)? I would like to consider the case I have a histogram showing residuals which are uniformly distributed between -1 and 1.
We still have errors that are symmetrically distributed around 0, so I would have thought that the estimates would still be the same as for the normally distributed errors. Is this true? What else can we say? What about the variances/p-values of the estimates?
 A: There are a few things we can say about this situation

*

*the condition of normality of residuals only needs to hold approximately. If the sample size is small then it can be difficult to distiguish a uniform from a normal distribution, and it is reasonable in such circumstances to assess the residuals as plausibly normal. With a large sample size this obviously is not the case.


*the estimates will be unbiased


*the estimates will be consistent


*the regression coefficient estimators will not be t distributed if using least squares, so the associated p values will be unreliable.
A: A classic linear regression model works under the assumption that the data can be modeled as
y = Ax + b + eta

where eta ~ N(0,sigma) .
if your residuals are uniformly distributed, it means that the above assumptions doesn't hold.
However, this linear regression can still work for you, depending on the application-  both a uniform and Gaussian model are symmetric, with E(data) = median(data). So even though the data is not really 'Gaussian', the line that best fit the data (according to the mean/median) will be the same (again, depending on the application).
An option that might work for you is Bayesian Linear Regression (BLR):
in BLR, you can choose your model assumptions: eta ~ U(-1,1), eta ~ N(0,sigma) or eta ~ Beta(2,2) are all valid assumptions. anything that best fit your data
