A hideous Linear Regression/confidence set question

[Phillips101]

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.hat-beta)' * X'*X * (beta.hat-beta), where t' is t transpose. I think I've done this. I think it's a sigma^2 chi-squared (n-p) distribution.

Next, Hence find a (1-a)-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 chi-squareds, 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

 

[statdad]

Notice that

$$\frac{\hat{\beta}' X'X \hat{\beta}}{\sigma^2}$$

has a $$\Chi^2$$ 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?

 

[Phillips101]

Use the MLE sigma2.hat=(1/n)*||Y-Xbeta.hat||^2 ? This is distributed as a chi-squared n-1 variable if I remember correctly...

 

[Phillips101]

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.hat-beta make any difference to this?

 

[blue_raver22]

I understand that this may seem like a daunting and confusing task, but I assure you that it is a crucial step in statistical analysis. Let's break down the problem and address each part separately.

First, let's start with the distribution of (beta.hat-beta)' * X'*X * (beta.hat-beta). This is known as the Hotelling's T-squared distribution, which is a multivariate extension of the chi-squared distribution. It follows a non-central chi-squared distribution with n-p degrees of freedom and a non-centrality parameter of (beta.hat-beta)' * X'*X * (beta.hat-beta)/sigma^2. This distribution can be used to calculate the confidence interval for beta.

Next, we need to find a (1-a)-level confidence set for beta. This is where the F distribution comes into play. The F distribution is used to test the equality of two variances, which in this case, is the ratio of (beta.hat-beta)' * X'*X * (beta.hat-beta)/sigma^2 divided by the residual sum of squares. This F statistic can be used to construct a confidence interval for beta using the Hotelling's T-squared distribution.

Since we are dealing with vectors, the confidence set for beta will be a region in the n-dimensional space. This region will be defined by a lower and upper bound for each component of beta. This can be visualized as a hyperellipsoid in the n-dimensional space.

Finally, to obtain the confidence set, we need to calculate the critical values for the F distribution and use them to construct the confidence interval for each component of beta. This will give us a region in the n-dimensional space where we can be (1-a)% confident that the true value of beta lies within.

In conclusion, constructing a confidence set for beta in a linear regression model may seem challenging, but it is a necessary step in statistical analysis. By understanding the underlying distributions and using appropriate statistical tests, we can obtain a region in the n-dimensional space where we can be confident that the true value of beta lies within. I hope this explanation has helped to clarify the process and I am happy to provide further assistance if needed.