CEBA talks 2020-2024
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Self-normalizing Tests Using the Cauchy Distribution

Authors: Mehdi Hosseinkouchack (EBS Business School, Econometrics Chair, EBS University)

Abstract: A testing principle is introduced where the statistic is the ratio of two weighted averages. Upon normalization and orthogonalization the ratio thus converges to the standard Cauchy distribution under the null hypothesis. At the same time a potential nuisance scaling parameter cancels from the ratio without having to be estimated, making these Cauchy tests self-normalizing (scale-invariant). These tests are not directed against specific alternatives but rather belong to the toolkit of general specification testing. Still, assuming specific local alternatives, asymptotic power can be determined since the ratio converges to a non-centered Cauchy distribution. Power crucially hinges on the weighting scheme when computing the weighted averages. We discuss three examples: tests for a Wiener process, for a Brownian bridge, and under long memory. Simulations show that the asymptotic scale-invariance is effective in finite samples, and the Cauchy test has better size control than competing procedures. This comes at the price that the Cauchy test is less powerful than competitors directed against specific alternatives. Finally, we indicate how Cauchy tests can be extended to a multivariate framework of correlated samples, such that the tests are robust with respect to cross-dependence without need to explicitly account for it.