*(English)*Zbl 0771.62033

Summary: Observations ${y}_{ij}$ are made at points $({x}_{1i},{x}_{2j})$ according to the model ${y}_{ij}=F({x}_{1i},{x}_{2j})+{e}_{ij}$, where the ${e}_{ij}$ are independent normals with constant variance. In order to test that $F({x}_{1},{x}_{2})$ is an additive function of ${x}_{1}$ and ${x}_{2}$, a likelihood ratio test is constructed comparing

where ${Z}_{1}$, ${Z}_{2}$ are Brownian motions and $Z$ is a Brownian sheet. The ratio of Brownian sheet variance to error variance $\alpha $ is chosen by maximum likelihood and the likelihood ratio test statistic $W$ of ${H}_{0}:\alpha =0$ 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.

##### MSC:

62G10 | Nonparametric hypothesis testing |

62E20 | Asymptotic distribution theory in statistics |

62M10 | Time series, auto-correlation, regression, etc. (statistics) |