Independence tests in the presence of measurement errors: An invariance law

In many scientific areas the observations are collected with measurement errors. We are interested in measuring and testing independence between random vectors which are subject to measurement errors. We modify the weight functions in the classic distance covariance such that, the modified distance covariance between the random vectors of primary interest is the same as its classic version between the surrogate random vectors, which is referred to as the invariance law in the present context. The presence of measurement errors may substantially weaken the degree of nonlinear dependence. An immediate issue arises: The classic distance correlation between the surrogate vectors cannot reach one even if the two random vectors of primary interest are exactly linearly dependent. To address this issue, we propose to estimate the distance variance using repeated measurements. We study the asymptotic properties of the modified distance correlation thoroughly. In addition, we demonstrate its finite-sample performance through extensive simulations and a real-world application.