Testing the goodness-of-fit of the stable distributions with applications to German stock index data and Bitcoin cryptocurrency data

Outlier-prone data sets are of immense interest in diverse areas including economics, finance, statistical physics, signal processing, telecommunications and so on. Stable laws (also known as \(\alpha \)- stable laws) are often found to be useful in modeling outlier-prone data containing important information and exhibiting heavy tailed phenomenon. In this article, an asymptotic distribution of a unbiased and consistent estimator of the stability index \(\alpha \) is proposed based on jackknife empirical likelihood (JEL) and adjusted JEL method. Next, using the sum-preserving property of stable random variables and exploiting U-statistic theory, we have developed a goodness-of-fit test procedure for \(\alpha \)-stable distributions where the stability index \(\alpha \) is specified. Extensive simulation studies are performed in order to assess the finite sample performance of the proposed test. Finally, two appealing real life data examples related to the daily closing price of German Stock Index and Bitcoin cryptocurrency are analysed in detail for illustration purposes.