Characterization-based approach for construction of goodness-of-fit test for L\'evy distribution

TL;DR

This paper introduces two new goodness-of-fit tests for the Le9vy distribution based on V-empirical Laplace transforms, demonstrating their effectiveness through theoretical properties, efficiency analysis, simulations, and real-data applications.

Contribution

The paper proposes two novel, scale-free goodness-of-fit tests for the Le9vy distribution using V-empirical Laplace transforms, extending existing methods and including a generalization of a recent test.

Findings

New tests are scale free under the null hypothesis.

The tests show high local Bahadur efficiency.

Empirical and real-data analyses confirm effectiveness.

Abstract

The L\'evy 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 L\'evy distribution. In this paper, two novel classes of goodness-of-fit tests for the L\'evy 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 L\'evy 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.