James Ankeny wrote:
>
> Hello,
> I have a question regarding the so-called "semi-studentized residual,"
> which is of the form (e_i)* = ( e_i - 0 ) / sqrt(MSE). Here, e_i is the ith
> residual, 0 is the mean of the residuals, and sqrt(MSE) means the square
> root of MSE. Now, if I understand correctly, the population simple linear
> regression model assumes that the E_i, the error terms, are independent and
> identically distributed random variables with N(0, sigma^2). My question is,
> are semi-studentized residuals not fully studentized because MSE is not the
> variance of all the residuals?
Correct. In fact, it probably isn't the variance of any of them, though
it
will often be reasonably close.
> It seems like MSE would be the variance of
> the residuals, unless of course the residuals from the sample data are not
> independent and identically distributed random variables.
Don't confuse errors with residuals. In the model, the error term
may be i.i.d., but the residuals (which estimate them) are neither
independent nor identically distributed.
> If not, each
> residual may have its own variance, in which case we would have to find this
> and studentize each residual by its own standard error? I am not sure if I
> am thinking about this in the right way.
> Also, if the E_i are iid random variables, does this mean that the
> observations Y_i are iid random variables within a particular level of X?
Yes.
Glen
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