its power can only compete with the power of the Wilcoxon test if the shift height is relatively high. Even though the self-normalized CUSUM test does not suffer from severe size distortions, its power cannot compete with the power of any other test. Obviously, CUSUM-based tests are highly unreliable when testing for a change in the variance. We will now investigate the finite sample performance of the CUSUM change-point tests for detecting changes in the tail parameter α of LMSV time series { X$_{j}$, j ≥ 1 } , i.e. we choose ψ ( x ) = log( | x | ) for the test statistics described in Section 3 and Section 5. In this case, the Hermite rank of Ψ( y ) = E ( ψ ( σ ( y ) ε$_{1}$ )) equals m = 1 and J$_{m}$ (Ψ) = 1 such that the limit of the CUSUM test statistic does not depend on α . We simulate observations X$_{1}$, . . . , X$_{n}$ which satisfy (16) for { ε$_{j}$, j ≥ 1 } that follow a (non-centered) Pareto distribution with parameter α . For the computation of the empirical power, we consider LMSV time series with a change-point of height h after a proportion τ of the data, i.e. we consider Pareto distributed random variables ε$_{j}$ , j = 1 , . . . , n with parameters α$_{j}$ , j = 1 , . . . , n such that α$_{j}$ = α for j = 1 , . . . , ⌊ nτ ⌋ while α$_{j}$ = α + h for j = ⌊ nτ ⌋ +1 , . . . , n . We computed the rejection rates of the testing procedures on the basis of 5 , 000 realizations of time series with sample sizes 500, 1 , 000 and 2 , 000. The simulation results for τ = 0 . 5 are reported in Table 10; additional simulation results for τ = 0 . 25 can be found in Section S2 of the supplementary material. The frequency of a type 1 error, i.e. the rejection rate under the hypothesis, corresponds to the values in the columns that are superscribed by “h = 0”. 6.3 Change in the tail 31