I have no intention to improve at this matter; however, I had devise a strategy based on the outputs of multi-tests on a single sample.
The Tests below, for Normality, were T1= Lilliefors, T3= Cramer - von Mises, the 2nd , testing Uniformity, is: L. Amaral Afonso, Pedro Duarte, Revue de Statistique Appliquée, tome 40, nº1, (1992), p77-79. ________________
Table 1- Cut-off values and right tail probability, Normal Data, when a triple test is performed
For example suppose a Neyman - Pearson test as this: H0: Normal against H1: Uniform. Given that, with N=60, the output frequencies are: __Normal Data: [1 0 0] = 0.526, [1 0 1] = 0.475 __Uniform : [1 0 0] = 1.000 The latter case: the 1st test always rejects normality, the 2nd and 3rd fails to reject.
We dare if we, testing a sample, observe [1 0 0] we can say that alpha, the probability to reject normality when testing normal data is as high as, 1.000 - 0.526 = 0.474.