Cauchy or not Cauchy? New goodness-of-fit tests for the Cauchy distribution

TL;DR

This paper introduces new goodness-of-fit tests for the Cauchy distribution based on a novel characterization, demonstrating their effectiveness through simulations and real data application.

Contribution

It presents a new characterization of the Cauchy distribution and develops a class of consistent goodness-of-fit tests with a Hilbert space framework.

Findings

Test is competitive with existing methods

Test is consistent against various alternatives

Applied successfully to cryptocurrency log-returns

Abstract

We introduce a new characterization of the Cauchy distribution and propose a class of goodness-of-fit tests to the Cauchy family. The limit distribution is derived in a Hilbert space framework under the null hypothesis and under fixed alternatives. The new tests are consistent against a large class of alternatives. A comparative Monte Carlo simulation study shows that the test is competitive to the state of the art procedures, and we apply the tests to log-returns of cryptocurrencies.