# More accurate, calibrated bootstrap confidence intervals for estimating the correlation between two time series

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## Abstract

Estimation of Pearson’s correlation coefficient between two time series, in the evaluation of the influences of one time-dependent variable on another, is an often used statistical method in climate sciences. Data properties common to climate time series, namely non-normal distributional shape, serial correlation, and small data sizes, call for advanced, robust methods to estimate accurate confidence intervals to support the correlation point estimate. Bootstrap confidence intervals are estimated in the Fortran 90 program PearsonT (Mudelsee, Math Geol 35(6):651–665, 2003), where the main intention is to obtain accurate confidence intervals for correlation coefficients between two time series by taking the serial dependence of the data-generating process into account. However, Monte Carlo experiments show that the coverage accuracy of the confidence intervals for smaller data sizes can be substantially improved. In the present paper, the existing program is adapted into a new version, called PearsonT3, by calibrating the confidence interval to increase the coverage accuracy. Calibration is a bootstrap resampling technique that performs a second bootstrap loop (it resamples from the bootstrap resamples). It offers, like the non-calibrated bootstrap confidence intervals, robustness against the data distribution. Pairwise moving block bootstrap resampling is used to preserve the serial dependence of both time series. The calibration is applied to standard error-based bootstrap Student’s \(t\) confidence intervals. The performance of the calibrated confidence interval is examined with Monte Carlo simulations and compared with the performance of confidence intervals without calibration. The coverage accuracy is evidently better for the calibrated confidence intervals where the coverage error is acceptably small already (i.e., within a few percentage points) for data sizes as small as 20.

## Keywords

Pearson’s correlation coefficient Bootstrap resampling Calibrated confidence interval Monte Carlo simulations Climate time series## Notes

### Acknowledgments

We thank Alexander Gluhovsky and three anonymous persons for constructive review comments. We thank Michael Schulz, Arne Biastoch, Jonathan Durgadoo, Frank Peeters, Conor Purcell, and Gema Martínez Méndez for discussions and helpful comments. The work described in this paper and the research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007–2013), Marie-Curie ITN, under Grant Agreement No. 238512, GATEWAYS project.

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