  Many statistical applications require the quantification of joint dependence
among more than two random vectors. In this work, we generalize the notion of
distance covariance to quantify joint dependence among d >= 2 random vectors.
We introduce the high order distance covariance to measure the so-called
Lancaster interaction dependence. The joint distance covariance is then defined
as a linear combination of pairwise distance covariances and their higher order
counterparts which together completely characterize mutual independence. We
further introduce some related concepts including the distance cumulant,
distance characteristic function, and rank-based distance covariance. Empirical
estimators are constructed based on certain Euclidean distances between sample
elements. We study the large sample properties of the estimators and propose a
bootstrap procedure to approximate their sampling distributions. The asymptotic
validity of the bootstrap procedure is justified under both the null and
alternative hypotheses. The new metrics are employed to perform model selection
in causal inference, which is based on the joint independence testing of the
residuals from the fitted structural equation models. The effectiveness of the
method is illustrated via both simulated and real datasets.
