Estimation of parameters of a normal distribution
Here we show that the knowledge of the distribution of for linear regression allows one to do without long calculations contained in the guide ST 2134 by J. Abdey.
Theorem. Let be independent observations from . 1) is distributed as 2) The estimators and are independent. 3) 4) 5) converges in distribution to
Proof. We can write where is distributed as Putting and (a vector of ones) we satisfy (1) and (2). Since we have Further,
Thus 1) and 2) follow from results for linear regression.
3) For a normal variable its moment generating function is (see Guide ST2133, 2021, p.88). For the standard normal we get
Applying the general property (same guide, p.84) we see that
4) By independence of standard normals
5) By standardizing we have and this converges in distribution to by the central limit theorem.