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

Abstract

Abstract 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.