Optimal linear Bayes and empirical Bayes estimation and prediction of the finite population mean

Chen and Shapiro (J. Statist. Comput. Simulation 53 (1995) 269) proposed the QH test for normality, based upon normalized spacings, which is easy to compute and has been shown by simulations to be as powerful as or superior to the original W-test. In this paper, we propose a generalized version of the W-type tests, named the weighted W-test which includes as special cases most versions of W-type tests. The limiting behavior of the weighted W statistics and the normalized version RH of the QH statistic are investigated. The relationship between QH and W is further discussed which interprets the underlying reason why the power property of the QH test is more likely to be that of the W test than that of the Wʹ test.