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Likelihood ratio test (LR-test) is frequently applied in regression analysis when the observations are normally distributed. However, when the normality assumption is violated, such a test can not be directly adopted in practice. In this work, we propose an approach by firstly transforming the observations to a set-indexed partial sums process. The limit model under this transformation is presented as a deterministic trend plus a random noise given by the set-indexed Brownian sheet. We show that the problem of testing the appropriateness of a model can be handled step-wise by testing the validity of the trend in the limit model by defining an LR-test based on the ratio between the likelihood function under  H 0  and under  H 0  ∪ H 1 . The rejection region as well as the power of the size  α  LR-test are obtained based on the Cameron-Martin-Girsanov formula of the Radon-Nikodym derivative of the set-indexed partial sums limit process of the observation with respect to the set-indexed Brownian sheet. Simulation study shows that the proposed test behaves as a consistent test in that it maximizes the power when  H 1  is true. Application of the method to real data can help in detecting valid regression model describing the variability of the maximum height of corn plants over the experimental region.

Likelihood ratio test for the mean of asymptotic spatial regression with the Brownian sheet noise