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Testing for additivity of a regression function. (English) Zbl 0771.62033
Summary: Observations $y\sb{ij}$ are made at points $(x\sb{1i},x\sb{2j})$ according to the model $y\sb{ij}=F(x\sb{1i},x\sb{2j})+e\sb{ij}$, where the $e\sb{ij}$ are independent normals with constant variance. In order to test that $F(x\sb 1,x\sb 2)$ is an additive function of $x\sb 1$ and $x\sb 2$, a likelihood ratio test is constructed comparing \align F(x\sb 1,x\sb 2) &=\mu+Z\sb 1(x\sb 1)+Z\sb 2(x\sb 2)\qquad\text{with}\\ F(x\sb 1,x\sb 2) &=\mu+Z\sb 1(x\sb 1)+Z\sb 2(x\sb 2)+Z(x\sb 1,x\sb 2),\endalign where $Z\sb 1$, $Z\sb 2$ 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 $H\sb 0: \alpha =0$ 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)
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