Many quantities have been discussed in the literature that can be used to characterise nonlinear time series. For the purpose of nonlinearity testing we need such quantities that are particular powerful in discriminating linear dynamics and weakly nonlinear signatures -- strong nonlinearity is usually more easily detectable. An important objective criterion that can be used to guide the preferred choice is the discrimination power of the resulting test. It is defined as the probability that the null hypothesis is rejected when it is indeed false. It will obviously depend on how and how strongly the data actually deviates from the null hypothesis.