Distribution functions of a spurious trend in a finite length data set with natural variability: Statistical considerations and a numerical experiment with a global circulation model

Abstract

A linear trend estimated from a finite‐length data set with random internal variability has a spurious component which is a difference from the true trend caused by changes in external conditions or parameters. Some moments and distribution functions of the spurious trend depending on the length of data are derived theoretically under general statistical assumptions. When the internal variability has a normal distribution, the spurious trend also has a normal distribution. In general cases of nonnormal distributions, we derive the distribution function of the spurious trend by the Edgeworth expansion. A few low‐order moments of the internal variability are necessary to obtain the approximate distribution function from the expansion. Population moments of the internal variability of a simple global circulation model are calculated using a 15,200‐year data set generated by a numerical experiment with a purely periodic annual forcing. Dependence of the estimation error of sample moments on the length of data is computed to evaluate an appropriate sample size for each moment. An ensemble experiment with the same model is used to estimate the detectability of a cooling trend in the stratosphere from a finite length data set with internal variability. Hypothesis tests for the statistical significance of the estimated trend are made: Student's t test, bootstrap test, and the more accurate test using the distribution function derived by the Edgeworth expansion. In the regions and seasons in which kurtosis of the internal variability is large the assumption that the spurious trend has a normal distribution is not appropriate, and the significance derived by the t test is different from that by the test using the Edgeworth expansion.