Exact Likelihood Ratio Test for the Parameters of the Linear Regression Model with Normal Errors

In this paper we present an exact likelihood ratio test (LRT) for testing the simple null hypothesis on all parameters of the linear regression model with normally distributed errors. In particular, we consider the simultaneous test for the regression parameters, β, and the error standard deviation, σ. The critical values of the LRT are presented for small sample sizes and a small number of explanatory variables for usual significance levels, α = 0.1, 0.05, and 0.01. The test is directly applicable for construction of the (1 - α)-confidence region for the parameters (β,σ) and the simultaneous tolerance intervals for future observations in linear regression models. For comparison, the suggested method for construction of the tolerance factors of the symmetric (1 - γ)-content simultaneous (1 - α)-tolerance intervals is illustrated by a simple numerical example.