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

Abstract

In this article we consider the linear regression model y = Xβ + ε, where ε 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 are the same as those from the hypothesis test H: λ = 0 versus K: λ ≥ 0, where λ is the expectation of the Lagrange multiplier arising from the estimation of β subject to the equality constraints Rβ = r. In this way we link the multivariate inequality constraints test to the much studied multivariate one-sided hypothesis test, H: μ = 0 versus K: μ ≥ 0, where μ is the mean of a multivariate normal random vector. We also show that the test H: R 1β ≥ r 1 and R 2β = r 2 versus K: β ∈ RK has the following equivalent test in terms of λ, H: λ = 0 versus K: λ1 ≥ 0, and λ2 ≠ 0, where λ1 is the subvector of λ corresponding to R1β ≥ r1 and λ2 corresponds to R 2β = r 2. Extensions of recent work in one-sided hypothesis testing for the coefficients of the linear regression model are also derived. For the normal linear regression model we derive exact tests for the hypothesis testing problems H: Rβ = r versus K: Rβ ≥ r and H: Rβ = r versus K: R 1β ≥ r, and R 2β ≠ r 2.