The use of empirical characteristic functions for inference problems, including estimation in some special parametric settings and testing for goodness of fit, has a long history dating back to the 70s (see for example, Feuerverger and Mureika (1977), and Csorgo (1981)). More recently, there has been renewed interest in using empirical characteristic functions in other inference settings. The distance covariance and correlation, developed by Szekely and Rizzo (2007) for measuring dependence and testing independence between two random vectors, are perhaps the best known illustrations of this. We apply these ideas to stationary univariate and multivariate time series to measure lagged auto- and cross-dependence in a time series. Assuming strong mixing, we establish the relevant asymptotic theory for the empirical auto- and cross-distance correlation functions. We also apply the auto-distance correlation function (adcf) to the residuals of an autoregressive processes as a test of goodness of fit. Under the null that an autoregressive model is true, the limit distribution of the empirical adcf can differ markedly from the corresponding one for an i.i.d. sequence. We illustrate the use of the empirical adcf for testing dependence and cross-dependence of time series in a variety of different contexts. This is joint work with Muneya Matsui (Nanzan University), Thomas Mikosch (University of Copenhagen) and Phyllis Wan (Columbia University).
Richard Davis is Chair and the Howard Levene Professor of Statistics at Columbia University. He is currently past-president of the Institute of Mathematical Statistics. He received his Ph.D. degree in Mathematics from the University of California at San Diego in 1979 and has held academic positions at MIT, Colorado State University, and visiting appointments at numerous other universities. Recently he was Hans Fischer Senior Fellow at the Technical University of Munich and Villum Kan Rasmussen Visiting Professor at the University of Copenhagen. Davis is a fellow of the Institute of Mathematical Statistics and the American Statistical Association, and is an elected member of the International Statistical Institute. He is co-author (with Peter Brockwell) of the bestselling books, "Time Series: Theory and Methods, 2nd edition", "Introduction to Time Series and Forecasting, 3rd edition", and the time series analysis computer software package, "ITSM2000". Together with Torben Andersen, Jens-Peter Kreiss, and Thomas Mikosch, he co-edited the "Handbook in Financial Time Series." In 1998, he won (with collaborator W.T.M Dunsmuir) the Koopmans Prize for Econometric Theory.
He has served on the editorial boards of major journals in probability and statistics and was Editor-in-Chief of the Bernoulli Journal, 2010-2012. He has advised/co-advised 32 PhD students and has presented short courses on time series and heavy-tailed modeling. His research interests include time series, applied probability, extreme value theory, and spatial-temporal modeling.