Self‐Normalising Tests Using the Cauchy Distribution

ABSTRACT A testing principle is introduced where the statistic is the ratio of two weighted averages. The ratio 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‐normalising. These tests are not directed against specific alternatives and belong to the toolkit of general specification testing. 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 the need to explicitly account for it.