How robust is the modified sequential triangular test of a correlation coefficient against non-normality of the basic variables?

There is a big lack of knowledge as concerns a test of the null hypothesis H₀: 0 < ρ ≤ ρ₀. Usually a test applies by some z-statistic according to Fisher (1921), which is approximately normally distributed. However, there is no evidence of whether the approximation is actually good enough—that is, it is of interest how the factual distribution of the respective test statistic holds the type-I risk—and which type-II risk results. Because this question cannot be answered theoretically at present, we try a simulation study in order to gain respective knowledge. For this, we even investigate the case of variables that are not (at all) normally distributed. Moreover, we consider variables not normally distributed but test the simple case of the exact t-test of H₀: ρ = ρ₀. The results show, in particular, that the test tracing back to R. A. Fisher does not hold the type-I risk if severe nonnormality of the variables’ distributions is given.