Bootstrap Estimation of Long-Range Dependence in Arfima Processes

Abstract

Long-memory or long-range dependence (LRD) denotes the property of a time series to exhibit a significant dependence between very distant observations. The presence of long-memory is established in diverse fields, for example, in hydrology, finance, network traffic, psychology, etc. From a practical point of view an important issue is the estimation of LRD parameters, especially the estimation of the Hurst parameter proposed in [19]. The Hurst parameter, usually referred as the H parameter, indicates the intensity of LRD.Various methods for estimating the H parameter in a time series data are available some of which are described in [3]. They are validated by appealing to an asymptotic analysis supposing that the sample size of the time series converges to infinity. Taqqu et al. [37] investigated empirical performance of nine methods for estimating the LRD in simulated autoregressive fractionally integrated moving average (ARFIMA) processes for a fairly large sample size of 10000 terms.The bootstrap is a general statistical procedure for statistical inference, originally introduced by Efron [11] that generally yields better estimations in finite sample sizes, than the classical methods. In this paper, we consider a bootstrap technique, which uses blocks composed of cycles, to estimate the H parameter in ARFIMA processes. The bias and the root mean square error (RMSE) of five estimators and their corresponded bootstrap estimators are obtained by a Monte Carlo study.