Constructing Locally Best Invariant Tests of the Linear Regression Model Using the Density Function of a Maximal Invariant

In the context of the linear regression model in which some regression coefficients are of interest and others are purely nuisance parameters, we derive the density function of a maximal invariant statistic after eliminating the nuisance parameters by the principle of invariance argument. This allows the construction of a range of optimal test statistics including the locally best invariant (LBI) test which is equivalent to the well-known one-sided t-test. The resultant LBI test is also found to be uniformly most powerful invariant (UMPI).