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Quantile-Based Inference for Tempered Stable Distributions


We introduce a simple, fast, and accurate way for the estimation of numerous distributions that belong to the class of tempered stable probability distributions. Estimation is based on the method of simulated quantiles (Dominicy and Veredas in J Econom 172:235–247, 2013). MSQ consists of matching empirical and theoretical functions of quantiles that are informative about the parameters of interest. In the Monte Carlo study we show that MSQ is significantly faster than maximum likelihood and the MSQ estimators can be nearly as precise as MLE’s. A Value at Risk study using 13 years of daily returns from 21 world-wide market indexes shows that the risk assessments of MSQ estimates are as good as MLE’s.

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Fig. 1


  1. The normal tempered stable distribution should not be confused with the normal distribution.

  2. Note that the accuracy of this approximation depends on the method of numerical integration. More detailed information can be found in Menn and Rachev (2006), Kim et al. (2010) and Fallahgoul et al. (2016).


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Hasan A. Fallahgoul gratefully acknowledges financial support from the Belgian Federal Science Policy Office (BELSPO) and the Marie Curie Actions from the European Commission and from the Swiss National Science Foundation Sinergia grant “Empirics of Financial Stablity” [154445]. We thank Davy Paindaveine, the editor, and two referees for insightful remarks.

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Correspondence to Frank J. Fabozzi.



See Tables 1, 2, 3, 4, 5, 6, 7 and 8.

Table 1 Monte Carlo study for the CTS distribution
Table 2 Monte Carlo study for the NTS distribution
Table 3 Monte Carlo study for the GTS distribution
Table 4 The speed of computation for MLE and MSQ
Table 5 Descriptive statistics of raw and filtered data
Table 6 Estimation results for the CTS distribution
Table 7 Estimation results for the NTS distribution
Table 8 VaR evaluation MLE versus MQS

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Fallahgoul, H.A., Veredas, D. & Fabozzi, F.J. Quantile-Based Inference for Tempered Stable Distributions. Comput Econ 53, 51–83 (2019).

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  • Heavy tailed distribution
  • Tempered stable distribution
  • Method of simulated quantiles

JEL classification

  • C5
  • G12