Testing for Additivity of a Regression Function

Observations yᵢⱼ are made at points (x₁ᵢ,x₂ⱼ) according to the model yiy=F(x₁ᵢ,x₂ⱼ)+eᵢⱼ, where the eᵢⱼ are independent normals with constant variance. In order to test that F(x₁,x₂) is an additive function of x₁ and x₂, a likelihood ratio test is constructed comparing F(x₁,x₂)=μ+Z₁ (x₁)+Z₂(x₂) with F(x₁,x₂)=μ+Z₁(x₁)+Z₂(x₂)+Z(x₁,x₂), where Z₁, Z₂ are Brownian motions and Z is a Brownian sheet. The ratio of Brownian sheet variance to error variance ∞ is chosen by maximum likelihood and the likelihood ratio test statistic W of H₀:∞=0 used to test for departures from additivity. The asymptotic null distribution of W is derived, and its finite sample size behaviour is compared with two standard tests in a simulation study. The W test performs well on the five alternatives considered.