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Abstract Linear regression is arguably the most widely used statistical method. With fixed regressors and correlated errors, the conventional wisdom is to modify the variance-covariance estimator to accommodate the known correlation structure of the errors. We depart from existing literature by showing that with random regressors, linear regression inference is robust to correlated errors with unknown correlation structure. The existing theoretical analyses for linear regression are no longer valid because even the asymptotic normality of the least squares coefficients breaks down in this regime. We first prove the asymptotic normality of the t statistics by establishing their Berry–Esseen bounds based on a novel probabilistic analysis of self-normalized statistics. We then study the local power of the corresponding t tests and show that, perhaps surprisingly, error correlation can even enhance power in the regime of weak signals. Overall, our results show that linear regression is applicable more broadly than the conventional theory suggests, and they further demonstrate the value of randomization for ensuring robustness of inference.
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Consistency of the Hill Estimator for Time Series Observed with Measurement Errors
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.
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- PAR ID:
- 10246705
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Journal of Time Series Analysis
- Volume:
- 41
- Issue:
- 3
- ISSN:
- 0143-9782
- Page Range / eLocation ID:
- p. 421-435
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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