## Abstract

Two common approaches for estimating a linear trend are 1) simple linear regression and 2) the epoch difference with possibly unequal epoch lengths. The epoch difference estimator for epochs of length M is defined as the difference between the average value over the last M time steps and the average value over the first M time steps divided by N - M, where N is the length of the time series. Both simple linear regression and the epoch difference are unbiased estimators for the trend; however, it is demonstrated that the variance of the linear regression estimator is always smaller than the variance of the epoch difference estimator for first-order autoregressive [AR(1)] time series with lag-1 autocorrelations less than about 0.85. It is further shown that under most circumstances if the epoch difference estimator is applied, the optimal epoch lengths are equal and approximately one-third the length of the time series. Additional results are given for the optimal epoch length at one end when the epoch length at the other end is constrained.

Original language | English (US) |
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Pages (from-to) | 9969-9976 |

Number of pages | 8 |

Journal | Journal of Climate |

Volume | 28 |

Issue number | 24 |

DOIs | |

State | Published - 2015 |

### Bibliographical note

Publisher Copyright:© 2015 American Meteorological Society.

## Keywords

- Mathematical and statistical techniques
- Regression analysis
- Statistical techniques
- Time series