Consistent testing of total independence based on the empirical characteristic function

Abstract

There are several tests for testing independence of two variables, but a shortage of tests that can be used to test total independence of several variables. The hypothesis of total independence Ho can be expressed in a simple manner in terms of the characteristic function, therefore the new test developed here is based on the empirical characteristic function. The new test statistic is an integral transformation of the empirical stochastic process constructed in accordance with the hypothesis of total independence. The asymptotic distributions of the test statistic under H₀ and under the alternative hypothesis H₁ are derived, when the weight function in the expression defining the test statistic satisfies certain conditions. The test is scale and location invariant and consistent. Also a nonparametric modification of the test for continuous variables is considered. The simulation study carried out here shows that the new tests, corresponding to two different weight functions, have almost as large empirical powers as the Blum-Kiefer-Rosenblatt test of independence for continuous data, where the dependence between the variables is linear. They have higher empirical powers, when the dependence is nonlinear. The new tests can also be applied for discrete data. The simulation study also shows that the weight function has some effect on the empirical powers. As an example we study the independence of variables in a data of functional capacity of retired women, and compare the results of the new tests to the model that is found by the GLIM-program, based on log-linear models. In another example we test the independence of estimated factor scores, which are derived by varimax-rotation from the Finnish ITPA data and from a data of school achievements of the pupils on 6th and 9th grade at the comprehensive schools in Jyväskylä.