Characterization-based approach for construction of goodness-of-fit test for Lévy distribution

Abstract

The Lévy distribution, alongside the Normal and Cauchy distributions, is one of the only three stable distributions whose density can be obtained in a closed form. However, there are only a few specific goodness-of-fit tests for the Lévy distribution. In this paper, two novel classes of goodness-of-fit tests for the Lévy distribution are proposed. Both tests are based on V-empirical Laplace transforms. New tests are scale free under the null hypothesis, which makes them suitable for testing the composite hypothesis. The finite sample and limiting properties of test statistics are obtained. In addition, a generalization of the recent Bhati–Kattumannil goodness-of-fit test to the Lévy distribution is considered. For assessing the quality of novel and competitor tests, the local Bahadur efficiencies are computed, and a wide power study is conducted. Both criteria clearly demonstrate the quality of the new tests. The applicability of the novel tests is demonstrated with two real-data examples.

Keywords:

• Bahadur efficiency
• Laplace transform
• stable distributions
• V-statistics

Mathematics Subject Classifications 2020:

• 62E10
• 62G10
• 62G20

Acknowledgments

We express our deep gratitude to professor Deepesh Bhati for providing us the code for computing empirical powers of JEL and AJEL test statistics. We are also grateful to two anonymous referees whose suggestions have contributed to the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of B. Milošević is supported by the Ministry of Science, Technological Development and Innovations of the Republic of Serbia (the contract 451-03-47/2023-01/ 200104). The work is also supported by the COST action CA21163 - Text, functional and other high-dimensional data in econometrics: New models, methods, applications (HiTEc).