Statistica Sinica 8(1998), 477-487

AN OPTIMAL TEST FOR THE MEAN FUNCTION

HYPOTHESIS

K. F. Cheng and J. W. Wu

National Central University and Tamkang University

Abstract: The conditional mean of the response variable Y ggiven the covariates is usually modelle d by a parametric function g(βx), where g(.) is a known function and β is a row vector of p unknown parameters. In this paper, a new method for testing the goodness of fit of the model g(βx) for the mean function is presented. The new test depends on the selection of weight functions. An expression for the efficacy of the proposed test under a sequence of local alternatives will be given. With the application of this result one can direct the choice of the optimal weight functions in order to maximize the efficacy. The new test is simple in computation and consistent against a broad class of alternatives. Asymptotically, the null distribution is independent of the underlying distribution of Y given X=x. Two pratical examples are given to illustrate the method. Further, simulation studies are given to show the advantages of the proposed test.

Key words and phrases: Consistency, estimation equation, goodness of fit, mean function, pitman efficiency.