Statistica Sinica 11(2001), 499-514 TEST OF HYPOTHESES BASED ON THE WEIGHTED LIKELIHOOD METHODOLOGY Claudio Agostinelli and Marianthi Markatou University of Padua and Columbia University Abstract: Weighted versions of the likelihood ratio, Wald, score and disparity tests are proposed for parametric inference. If the parametric model is correct, the weighted likelihood tests are asymptotically equivalent to the corresponding likeli- hood based tests, while the disparity test has asymptotically the same distribution as that of ∑ p i=1 λi Z 2 i , where Zi are standard normal random variables and λi are eigenvalues of an appropriate matrix. The tests have high level and power break- down points and they perform well in finite samples. A simulation study and a data example illustrate the performance of the tests in the presence of symmetric and asymmetric contamination. Key words and phrases: Asymptotic distribution, breakdown function, likelihood ratio test, robustness, score test, Wald test, weighted likelihood. 1. Introduction The investigation of robustness of testing procedures dates back to 1931, when E. S. Pearson discovered the nonrobustness of the test of variances. How- ever, most research effort has focused on robust estimation and less attention has been given to tests. The aim of robust testing is twofold. The level of the test should be stable under small, arbitrary departures from the null hypothesis (robustness of valid- ity), and the test should retain good power under small, arbitrary departures from specified alternatives (robustness of efficiency). Many authors have pro- posed tests with desirable robustness properties. Those include Huber (1965, 1981), Schrader and Hettmansperger (1980), Ronchetti (1982), Markatou and Hettmansperger (1990), Markatou and He (1994), Heritier and Ronchetti (1994), He (1991) and Simpson (1989), to mention just a few. For a review, see Markatou, Stahel and Ronchetti (1991). In this article, we present robust versions of the Wald, score, and likelihood ratio-type tests that are based on the weighted likelihood estimators (WLEE) proposed by Markatou, Basu and Lindsay (1997, 1998). Section 2 gives a short review of the weighted likelihood methodology. Section 3 presents the test statis- tics under study and looks at disparity test analogous to the one studied by