Estimation of long-range dependence in gappy Gaussian time series

Abstract

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 noise, 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.

Estimation of Long-Range Dependent Models with Missing Data: to Input or not to Input?

“…In this work, we consider three different native estimators for d. The semiparametric estimators proposed by Knight et al (2017) and Craigmile and Mondal (2020) are both wavelet-based, relying on a relationship between the undecimated wavelet variance and the parameter d. The former estimates the wavelet variance using wavelet lifting while the latter uses a specially designed estimator. The third method was proposed by Pumi et al (2023), which is the only copula-based estimator for the long-range parameter d for univariate time series in the literature.…”

“…for large j and some constant C. Given an estimator of v 2 j , v2 j for j ∈ {j 0 , • • • , j 0 + m} for positive j 0 and m, d can be estimated by regressing log(v 2 j 0 ), • • • , log(v 2 j 0 +m ) in j 0 , • • • , j 0+m . The main contribution in Craigmile and Mondal (2020) is to propose an unbiased, consistent, and asymptotically normally distributed estimator for v 2 j in long-range dependent processes containing missing data. However, to estimate d from v2 j , using a regression approach will depend on an unknown dispersion matrix.…”

“…A second estimator is also inspired by the work of Abry and Veitch (1998) for complete long-range dependent processes, using only the diagonal elements of the full dispersion matrix to estimate d. The authors called it the diagonal estimator (called C.abry here). More details can be found in Craigmile and Mondal (2020).…”

“…The argument R in the code was kept at its default value of 7. More information can be found in Craigmile and Mondal (2020).…”

“…Another wavelet-based estimator for the long memory parameter d is presented in Knight et al (2017). The method is based on a multiscale lifting transformation known as LOCAAT (lifting one coefficient at a time) proposed by Jansen et al (2001) and it is similar in nature to Craigmile and Mondal (2020)'s method. The LoMPE's idea is to apply the LOOCAT method to obtain a collection of lifting coefficients, which, after suitable normalization, are used to estimate the wavelet's coefficient variance.…”