*2.3. Look-Up Table*

To perform this hypothesis test on a given set of data, we firstly estimate the heaviness of tail using Hill's estimator *h*Hill. Then, the null distribution of concavity measures is constructed by

<sup>5</sup> Knots were chosen so that the function is evaluated on 18 equal sub-intervals. This gave the best approximation considering computational efficiency.

simulating from Student *t*-distributed processes corresponding to tail indices ranging over the interval [*h*Hill − 0.5, *h*Hill + 0.5]. A larger range of tail index is used to allow for variation in our tail index estimation, while allowing more observations for constructing the concavity measure distribution.<sup>6</sup> The 5th percentile of this distribution is the critical value for the hypothesis test.

The null hypothesis is rejected if the observed test statistic is less than the calculated critical value. To carry out the hypothesis test, a look-up table is generated to summarise the tail indices and the corresponding critical values based on both the localised and global concavity measures. A simplified version of the look-up table is presented in Appendix A. The critical values are presented for tail indices 0.5, 1.0, 1.5, ..., with the largest tail index being 10.
