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Testing for additivity of a regression function. (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

F(x 1 ,x 2 )=μ+Z 1 (x 1 )+Z 2 (x 2 )withF(x 1 ,x 2 )=μ+Z 1 (x 1 )+Z 2 (x 2 )+Z(x 1 ,x 2 ),

where Z 1 , Z 2 are Brownian motions and Z is a Brownian sheet. The ratio of Brownian sheet variance to error variance α is chosen by maximum likelihood and the likelihood ratio test statistic W of H 0 :α=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:
62G10Nonparametric hypothesis testing
62E20Asymptotic distribution theory in statistics
62M10Time series, auto-correlation, regression, etc. (statistics)