**9. Findings and Concluding Remarks**

It can be observed from Table 1 that the asymptotic variance of the untruncated estimator is reduced for the corresponding truncated estimator, indicating the efficiency of the truncated estimator.

It can also be noted from Table 2 that, for *α* = 1.01, the RMSE of the modified truncated estimator is less than that of the Hill estimator when the sample is relocated by three different relocations, viz. true mean = 0, sample mean, and sample median, for higher values of the concentration parameter *ρ* = 0.5, 0.6, 0.8, and 0.9 for sample sizes n = 100, 250, 500, and 1000 and for *ρ* = 0.3, 0.4, 0.6, 0.8, and 0.9 for sample sizes n = 2000, 5000, and 10,000. Furthermore, it can be observed that, for *α* = 1.25, 1.5, 1.75 and 1.9, the RMSE of the modified truncated is less than that of the Hill estimator for different relocations for *ρ* = 0.6, 0.7, 0.8, and 0.9 for smaller sample size and even for *ρ* = 0.5 for larger sample size. This clearly indicates the efficiency of the modified truncated estimator over the Hill estimator for higher values of the concentration parameter *ρ*.

It can be observed in Table 3 that the RMSE of the modified truncated estimator is less than that of the characteristic function-based estimator for almost all values of *α* corresponding to all values of *σ*.

The Hill estimator (Dufour and Kurz-Kim (2010)) is defined for 1 ≤ *α* ≤ 2, whereas the modified truncated estimator is defined for the whole range 0 ≤ *α* ≤ 2. In addition, the overall performance of the modified truncated estimator is quite good in terms of efficiency and consistency over both the Hill estimator and the characteristic function-based estimator.

Thus, we have established an estimator of the index parameter *α* that strongly supports its parameter space (0, 2]. It can be observed from the above real life data applications that the modified truncated estimator is quite close to that of the characteristic function-based estimator. In addition, it is simpler and computationally easier than that of the estimator defined in Anderson and Arnold (1993). Thus, it may be considered as a better estimator.

Again, when the estimator of *α* lies between 1 and 2, is attempted to model a mixture of two distributions with the value of the index parameter as that of the two extreme tails that is modeling a mixture of Cauchy (*α* = 1) and normal (*α* = 2) distributions when 1 < *α* < 2 or modeling a mixture of Double Exponential (*α* = 1 2 ) and Cauchy (*α* = 1) distributions when 1 2 < *α* < 1. Then, it is compared with that of the stable family of distributions for goodness of fit.

We could have used the usual technique of non-linear optimization as used in Salimi et al. (2018) for estimation, but it is computationally demanding and also the (statistical) consistency of the estimators obtained by such method is unknown. In contrast, our proposed methods of trigonometric moment and modified truncated estimation are much simpler, computationally easier and also possess useful consistency properties and, even their asymptotic distributions can be presented in simple and elegant forms as already proved above.

**Author Contributions:** Problem formulation, formal analyses and data curation, A.S.; formal and numerical data analyses, M.R.

**Acknowledgments:** The research of the second author of the paper was funded by the Senior Research Fellowship from the University Grants Commission, Government of India. She is also thankful to the Indian Statistical Institute and the University of Calcutta for providing the necessary facilities.

**Conflicts of Interest:** The authors declare no conflicts of interest.
