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A hideous Linear Regression/confidence set question 
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#1
Apr110, 08:21 AM

P: 33

Take the linear model Y=X*beta+e, where e~Nn(0, sigma^2 * I), and it has MLE beta.hat
First, find the distribution of (beta.hatbeta)' * X'*X * (beta.hatbeta), where t' is t transpose. I think I've done this. I think it's a sigma^2 chisquared (np) distribution. Next, Hence find a (1a)level confidence set for beta based on a root with an F distribution. I can't do this to save my life. I'm aware that an F distribution is the ratio of two chisquareds, but where the hell I'm going to get another chi squared from I have no idea. Also, we're dealing in vectors and I don't know how,what,why any confidence set is going to be or even look like, and I've no idea how to even try to get one. Any help would be appreciated. Thanks 


#2
Apr210, 07:08 PM

HW Helper
P: 1,361

Notice that
[tex] \frac{\hat{\beta}' X'X \hat{\beta}}{\sigma^2} [/tex] has a [tex] \Chi^2 [/tex] distribution. however, the variance is unknown, so you need to estimate it (with another expression from the regression). What would you use for the estimate, and what is its distribution? 


#3
Apr310, 05:43 AM

P: 33

Use the MLE sigma2.hat=(1/n)*YXbeta.hat^2 ? This is distributed as a chisquared n1 variable if I remember correctly...



#4
Apr310, 05:45 AM

P: 33

A hideous Linear Regression/confidence set question
If that's correct, then the thing you posted is distributed as an F distribution, which is what I need? And would swapping beta.hat for beta.hatbeta make any difference to this?



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