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 rankbased 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.where f X , f Y and f X,Y are the individual and joint characteristic functions of X and Y respectively,being the complete gamma function. An important feature of dCov is that it fully characterizes independence because dCov(X, Y ) = 0 if and only if X and Y are independent. Many statistical applications require the quantification of joint dependence among d ≥ 2 random variables (or vectors). Examples include model diagnostic checking for directed acyclic graph (DAG)where inferring pairwise independence is not enough in this case (see more details in Section 6), and independent component analysis which is a means for finding a suitable representation of multivariate data such that the components of the transformed data are mutually independent. In this paper, we shall introduce new metrics which generalize the notion of dCov to quantify joint dependence of d ≥ 2 random vectors. We first introduce the notion of high order dCov to measure the so-called Lancaster interaction dependence (Lancaster, 1969). We generalize the notion of Brownian covariance (Székely et al., 2009) and show that it coincides with the high order distance covariance. We then define the joint dCov (Jdcov) as a linear combination of pairwise dCov and their high order counterparts. The proposed metric provides a natural decomposition of joint dependence into the sum of lower order and high order effects, where the relative importance of the lower order effect terms and the high order effect terms is determined by a user-chosen number. In the population case, Jdcov is equal to zero if and only if the d random vectors are mutually independent, and thus completely characterizes joint independence. It is also worth mentioning that the proposed metrics are invariant to permu...

The paper presents new metrics to quantify and test for (i) the equality of distributions and (ii) the independence between two high-dimensional random vectors. We show that the energy distance based on the usual Euclidean distance cannot completely characterize the homogeneity of two highdimensional distributions in the sense that it only detects the equality of means and the traces of covariance matrices in the high-dimensional setup. We propose a new class of metrics which inherits the desirable properties of the energy distance and maximum mean discrepancy/(generalized) distance covariance and the Hilbert-Schmidt Independence Criterion in the low-dimensional setting and is capable of detecting the homogeneity of/completely characterizing independence between the low-dimensional marginal distributions in the high dimensional setup. We further propose t-tests based on the new metrics to perform high-dimensional two-sample testing/independence testing and study their asymptotic behavior under both high dimension low sample size (HDLSS) and high dimension medium sample size (HDMSS) setups. The computational complexity of the t-tests only grows linearly with the dimension and thus is scalable to very high dimensional data. We demonstrate the superior power behavior of the proposed tests for homogeneity of distributions and independence via both simulated and real datasets.

The paper presents new metrics to quantify and test for (i) the equality of distributions and (ii) the independence between two highdimensional random vectors. We show that the energy distance based on the usual Euclidean distance cannot completely characterize the homogeneity of two high-dimensional distributions in the sense that it only detects the equality of means and the traces of covariance matrices in the highdimensional setup. We propose a new class of metrics which inherits the desirable properties of the energy distance and maximum mean discrepancy/(generalized) distance covariance and the Hilbert-Schmidt Independence Criterion in the low-dimensional setting and is capable of detecting the homogeneity of/completely characterizing independence between the low-dimensional marginal distributions in the high dimensional setup. We further propose t-tests based on the new metrics to perform highdimensional two-sample testing/independence testing and study their asymptotic behavior under both high dimension low sample size (HDLSS) and high dimension medium sample size (HDMSS) setups. The computational complexity of the t-tests only grows linearly with the dimension and thus is scalable to very high dimensional data. We demonstrate the superior power behavior of the proposed tests for homogeneity of distributions and independence via both simulated and real datasets.

Change-point detection has been a classical problem in statistics and econometrics. This work focuses on the problem of detecting abrupt distributional changes in the data-generating distribution of a sequence of high-dimensional observations, beyond the first two moments. This has remained a substantially less explored problem in the existing literature, especially in the high-dimensional context, compared to detecting changes in the mean or the covariance structure.We develop a nonparametric methodology to (i) detect an unknown number of change-points in an independent sequence of high-dimensional observations and (ii) test for the significance of the estimated change-point locations. Our approach essentially rests upon nonparametric tests for the homogeneity of two high-dimensional distributions. We construct a single change-point location estimator via defining a cumulative sum process in an embedded Hilbert space. As the key theoretical innovation, we rigorously derive its limiting distribution under the high dimension medium sample size (HDMSS) framework. Subsequently we combine our statistic with the idea of wild binary segmentation to recursively estimate and test for multiple change-point locations.The superior performance of our methodology compared to other existing procedures is illustrated via extensive simulation studies as well as over stock prices data observed during the period of the Great Recession in the United States.

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