Distribution of the estimator of the error variance
If you are reading the book by Dougherty: this post is about the distribution of the estimator defined in Chapter 3.
where the deterministic matrix is of size satisfies (regressors are not collinear) and the error satisfies
is estimated by Denote Using (1) we see that and the residual is estimated by
is a projector and has properties which are derived from those of
If is an eigenvalue of then multiplying by and using the fact that we get Hence eigenvalues of can be only or The equation
tells us that the number of eigenvalues equal to 1 is and the remaining are zeros. Let be the diagonal representation of Here is an orthogonal matrix,
and is a diagonal matrix with eigenvalues of on the main diagonal. We can assume that the first numbers on the diagonal of are ones and the others are zeros.
Theorem. Let be normal. 1) is distributed as 2) The estimators and are independent.
Proof. 1) We have by (4)
Denote From (2) and (5)
and is normal as a linear transformation of a normal vector. It follows that where is a standard normal vector with independent standard normal coordinates Hence, (6) implies
(3) and (7) prove the first statement.
2) First we note that the vectors are independent. Since they are normal, their independence follows from
It's easy to see that This allows us to show that is a function of :
Independence of leads to independence of their functions and