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

• Published in: Journal of time series analysis Vol. 41; no. 3; pp. 421 - 435
• Main Authors: Kim, Mihyun, Kokoszka, Piotr
• Format: Journal Article
• Language: English
• Published: Oxford, UK John Wiley & Sons, Ltd 01-05-2020; Blackwell Publishing Ltd
• Subjects: Anomalies; Asymptotic methods; Asymptotic properties; Computer simulation; Discrete time; Hill estimator; Justification; Measurement; measurement error; Monte Carlo simulation; regular variation; Robustness; Robustness (mathematics); Roundoff error; Simulation; Time measurement; Time series; United States > US
• ISSN: 0143-9782; 1467-9892
• DOI: 10.1111/jtsa.12515

We investigate the asymptotic and finite sample behavior of the Hill estimator applied to time series contaminated by measurement or other errors. We show that for all discrete time models used in practice, whose non‐contaminated 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 computed with a roundoff error.