19.2 Linear regression models with Gaussian dependent variables
Assuming Gaussian distributions for the dependent variables Yi = Yxi the estimators from the Gauss–Markov theorem have additional properties. These can be used to construct confidence intervals for the parameters. In order to obtain the confidence intervals some results are necessary. These are given in the following theorems.
Assuming a linear regression model (19.4) and all conditions from Theorem 19.1 and additionally the stochastic quantities Y1,…,Yn to have Gaussian distributions, i.e. has multivariate Gaussian distribution where In is the n × n identity matrix
then the following holds:
1. The estimator is also the maximum likelihood estimator for θ.
2. The distribution of the estimator is a multivariate normal distribution
3. The pivotal quantity has chi-square distribution with n − k − 1 degrees of ...