Estimation of long-range dependence in gappy Gaussian time series

Knowledge of the long-range dependence (LRD) parameter is critical to studies of self-similar behavior. However, statistical estimation of the LRD parameter becomes difficult when the observed data are masked by short-range dependence and other noises or are gappy in nature (i.e., some values are missing in an otherwise regular sampling). Currently there is a lack of theory for spectral- and wavelet-based estimators of the LRD parameter for gappy data. To address this, we estimate the LRD parameter for gappy Gaussian semiparametric time series based upon undecimated wavelet variances. We develop estimation methods by using novel estimators of the wavelet variances, providing asymptotic theory for the joint distribution of the wavelet variances and our estimator of the LRD parameter. We introduce sandwich estimators to compute standard errors for our estimates. We demonstrate the efficacy of our methods using Monte Carlo simulations and provide guidance on practical issues such as how to select the range of wavelet scales. We demonstrate the methodology using two applications: one for gappy Arctic sea-ice draft data and another for gap-free and gappy daily average temperature data collected at 17 locations in south central Sweden.