Abstract
We establish asymptotic and finite sample properties of the Hill and Harmonic Moment estimators applied to heavy-tailed data contaminated by errors. We formulate conditions on the errors and the number of upper order statistics under which these estimators continue to be asymptotically normal. We specify analogous conditions which must hold in finite samples for the confidence intervals derived from the asymptotic normal distribution to be reliable. In the large sample analysis, we specify conditions related to second-order regular variation and divergence rates for the number of upper order statistics, k, used to compute the estimators. In the finite sample analysis, we examine several data-driven methods of selecting k, and determine which of them are most suitable for confidence interval inference. The results of these investigations are applied to interarrival times of internet traffic anomalies, which are available only with a round-off error.
Acknowledgments
Section 4 uses a proprietary data product derived from historical US-wide internet traffic measurements. We thank Professor Anura P. Jayasumana of Colorado State University's Department of Electrical and Computer Engineering for making it available to us.
Disclosure statement
No potential conflict of interest was reported by the author(s).