An Omnibus Test for Departures from Constant Mean

Observations yᵢ are made at points xᵢ according to the model yᵢ = F(xᵢ) + eᵢ, where the eᵢ are independent normals with constant variance. In order to decide whether or not $F(x)$ is constant, a likelihood ratio test is constructed, comparing F(x) ≡ μ with F(x) = μ + Z(x), where $Z(x)$ is a Brownian motion. The ratio of error variance to Brownian motion variance is chosen to maximize the likelihood, and the resulting maximum likelihood statistic B is used to test departures from constant mean. Its asymptotic distribution is derived and its finite sample size behavior is compared with five other tests. The B-statistic is comparable or superior to each of the tests on the five alternatives considered.