Asymptotic and finite sample properties of Hill-type estimators in the presence of errors in observations

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.

Keywords:

• Asymptotic normality
• harmonic moment estimator
• Hill estimator
• measurement error
• regular variation

2010 Mathematics Subject Classifications:

• 62F12
• 62P30

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).

Additional information

Funding

This research has been partially supported by the National Science Foundation grant DMS-1737795: ATD: Spatio-Temporal Model for the Propagation of Internet Traffic Anomalies.