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Abstract

A task in statistics is to find meaningful associations or dependencies between multivariate random variables or in multivariate, time-dependent stochastic processes. Hawkes (1971) introduced the powerful multivariate point process model of mutually exciting processes (Hawkes model) to explain causal structure in data. Therefore, we discuss several causality concepts and show that causal structure is fully encoded in the corresponding Hawkes kernels. Hence, for causal inference and for establishing graphical models induced by causality it is necessary to estimate the Hawkes kernels. We provide a nonparametric, consistent and asymptotically normal estimator of the Hawkes kernels depending on the increments on a time scale with mesh $\Delta$ using methods from infinite order regression and time series analysis. To illustrate our results we apply our method to EEG data from the spinal dorsal horn of a rat.

To tackle the problem for random samples of random vectors we examine a new dependence measure, namely distance correlation (Sz\'ekely, Rizzo and Bakirov; 2007). Distance correlation provides a strikingly simple sample version in order to test for independence between two random vectors of arbitrary dimensions and finite first moments. However, distance correlation is not well understood on the population side and it fails to be invariant under the group of all invertible affine transformations. Hence, we introduce the affinely invariant distance correlation and compute the analytic usual distance correlation and affinely invariant distance correlation in various settings: for multivariate normal distributions and for Lancaster probabilities (e.g. the bivariate gamma distribution) explicitly. Furthermore, we generalize an integral which is at the core of distance correlation.