Analytical investigation of the bias effect in variance-type estimators for inference of long-range dependence

Since the publication of the Bellcore measurements in the early nineties, long-range dependence (LRD) has been in the center of a continuous debate within the teletraac community. While researchers largely acknowledge the signiicance of the LRD phenomenon, they still disagree on two issues: (1) the utility of LRD models in buuer dimensioning and bandwidth allocation, and (2) the ability of commonly used statistical tools to diierentiate between true LRD and other potential interpretations of it (e.g., non-stationarity). This paper is related to the second issue. More speciically, our objective is to analytically demonstrate the limitations of variance-type indicators of LRD. Our work is not meant to advocate a particular modeling philosophy (be it LRD or SRD), but rather to emphasize the potential misidentiication caused by the inherent bias in variance-type estimators. Such misidentiication could lead one to wrongly conclude the presence of an LRD structure in a sequence that is known to be SRD. Our approach is based on deriving simple analytical expressions for the slope of the aggregated variance in three autocorrelated traac models: a class of SRD (but non-Markovian) M=G=1 processes, the discrete autoregressive of order one model (SRD Markovian), and the fractional ARIMA process (LRD). Our main result is that a variance-type estimator often indicates, falsely, the existence of an LRD structure (i.e., H > 0:5) in synthetically generated traces from the two SRD models. The bias in this estimator, however, diminishes monotonically with the length of the trace. We provide some guidelines on selecting the minimum trace length so that the bias is negligible. We also contrast the VT estimator with three other estimation techniques.