**7. Computational Studies**

The analytical variance of the untruncated moment estimator was compared with that of the modified truncated estimator, as presented in Table 1, for values of *α* < 1, which is more applicable in practical situations for volatile data.

The comparison of the performances of the two estimators is shown in Table 2. The parameter configurations were chosen as given by Hill (1975) and Dufour and Kurz-Kim (2010). The simulation is presented in Table 2 for the values of *α* = 1.01, 1.25, 1.5, 1.75, and 1.9 each with sample size n = 100, 250, 500, 1000, 2000, 5000, and 10,000 and for different values of *ρ* = 0.2, 0.4, 0.6, and 0.8 when skewness parameter *β* = 0, location parameter *μ* = 0, and scale parameter *σ* = (− ln(*ρ*))(1/*α*), i.e., concentration parameter *ρ* = *e*<sup>−</sup>*σ<sup>α</sup>* . For each combination of *α* and n, 10,000 replications were performed. In this simulation, the sample was relocated by three different relocations, viz. true mean = 0, estimated sample mean, and estimated sample median, and comparison of the root mean square errors (RMSEs) was made.

Next, in Table 3, comparison of the performance of the modified truncated estimator *α*ˆ2 with that of the characteristic function-based estimator where the simulation is presented for the values of *α* = 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, and 2.0 each with sample size n = 20, 30, 40, and 50, and the values of *σ* were taken as 3, 5, and 10 . For each combination of *α* and n, 10,000 replications were performed.


**Table 3.** Comparison of the RMSEs of the modified truncated estimator *α*ˆ3 (RMSE3) and the characteristic function-based estimator (RMSE4) when *μ* = 0 and *σ* unknown.


**Table 3.** *Cont.*

The asymptotic variance of the characteristic function-based estimator, unlike that of the modified truncated estimator, is not available in any closed analytical form. We are thus unable to present the Asymptotic Relative Efficiency (ARE) of these estimators of *α* analytically. Instead, we compared these through their MSEs based on extensive simulations over all reasonable small, moderate, and large sample sizes.
