Abstract

Knowledge of the long range dependence (LRD) parameter is critical to studies of fractal and self-similar behavior. However, statistical estimation of the LRD parameter becomes difficult when observed data are masked by short range dependence and other noise, or are gappy in nature (i.e., some values are missing in an otherwise regular sampling). We investigate estimation of the LRD parameter for Gaussian time series based upon wavelet variances. In the non-gappy case, our least-squares-based approach extends and improves upon existing methods by incorporating correlations between wavelet scales. For the more difficult gappy case, we also develop estimation methods by using novel estimators of the wavelet variances. In each case, we provide asymptotic theory and introduce sandwich estimators to compute the standard errors. Using Monte Carlo simulations, we highlight the improvements that are possible over existing approaches, and provide guidance on practical issues such as how to select the range of wavelet scales. We consider two applications; one for gappy Arctic sea-ice draft data, and another for non-gappy and gappy daily average temperature data collected at 17 locations in south central Sweden.

This research project is joint with Debashis Mondal, Ph.D., at the University of Chicago.