Consistency of the Hill Estimator for Time Series Observed with Measurement Errors

• Activity Number: 512 - Predicting and Evaluating Risk Models Within Distributions and Across Time
• Type: Contributed
• Date/Time: Wednesday, July 31, 2019 : 10:30 AM to 12:20 PM
• Sponsor: Section on Risk Analysis
• Author(s): Mihyun Kim* and Piotr Kokoszka
• Companies: Colorado State University and Colorado State University
• Keywords: Hill estimator; Regular variation; Measurement error

Abstract

We investigate the asymptotic and finite sample behavior of the Hill estimator applied to data contaminated by measurement or other errors. We show that for all discrete time stochastic models used in practice, whose marginal distributions are regularly varying, the Hill estimator is consistent. Essentially, the only assumption on the errors is that they have lighter tails than the underlying unobservable process. The asymptotic justification however depends on the specific class of models assumed for the underlying unobservable process. We show by means of a simulation study that the asymptotic robustness of the Hill estimator is clearly manifested in finite samples. We further illustrate this robustness by a numerical study of the interarrival times of anomalies in a backbone internet network, the Internet2 in the United States; the anomalies arrival times are measured with a roundoff error.