ASYMPTOTICALLY OPTIMAL TESTS AND OPTIMAL DESIGNS FOR TESTING THE MEAN IN REGRESSION MODELS WITH

APPLICATIONS TO CHANGE-POINT PROBLEMS

WOLFGANG BISCHOFF AND FRANK MILLER

Institute of Mathematical Stochastics, Department of Mathematics, University of Karlsruhe, D-76128 Karlsruhe, Germany

(Received July 27, 1998; revised July 2, 1999)

Abstract. Let a linear regression model be given with an experimental region [a, b] ⊆ R and regression functions f_{1}, …, f_{d+1} : [a, b] → R. In practice it is an important question whether a certain regression function f_{d+1}, say, does or does not belong to the model. Therefore, we investigate the test problem H_{0} : “f_{d+1} does not belong to the model” against K : “f_{d+1} belongs to the model” based on the least-squares residuals of the observations made at design points of the experimental region [a, b]. By a new functional central limit theorem given in Bischoff (1998, Ann. Statist., 26, 1398–1410), we are able to determine optimal tests in an asymptotic way. Moreover, we introduce the problem of experimental design for the optimal test statistics. Further, we compare the asymptotically optimal test with the likelihood ratio test (F-test) under the assumption that the error is normally distributed. Finally, we consider real change-point problems as examples and investigate by simulations the behavior of the asymptotic test for finite sample sizes. We determine optimal designs for these examples.

Key words and phrases: Asymptotically optimal tests, linear regression, F-test, likelihood ratio test, Gaussian processes, optimal designs, change-point problem, quality control.