Summary: Observations are made at points according to the model , where the are independent normals with constant variance. In order to test that is an additive function of and , a likelihood ratio test is constructed comparing
where , are Brownian motions and is a Brownian sheet. The ratio of Brownian sheet variance to error variance is chosen by maximum likelihood and the likelihood ratio test statistic of used to test for departures from additivity.
The asymptotic null distribution of is derived, and its finite sample size behaviour is compared with two standard tests in a simulation study. The test performs well on the five alternatives considered.