An Exact Test for Multiple Inequality and Equality Constraints in the Linear Regression Model

Abstract

Abstract In this article we consider the linear regression model y = Xβ + e, where e is N(0, σ2I). In this context we derive exact tests of the form H: Rβ ≥ r versus K: β ∈ RK for the case in which θ2 is unknown. We extend these results to consider hypothesis tests of the form H: R1β ≥ r1 and R2β = r2 versus K: (β ∈ RK . For each of these hypotheses tests we derive several equivalent forms of the test statistics using the duality theory of the quadratic programming. For both tests we derive their exact distribution as a weighted sum of Snedecor's F distributions normalized by the numerator degrees of freedom of each F distribution of the sum. A methodology for computing critical values as well as probability values for the tests is discussed. The relationship between this testing framework and the multivariate one-sided hypothesis testing literature is also discussed. In this context we show that for any size of the hypothesis test H: λ = 0 versus K: β ∈ RK the test statistic and critical value obtained a...