Some moments and distribution functions of the spurious trend are derived theoretically under general statistical assumptions. The moments of odd order are 0, and the standard deviation and kurtosis become smaller as the length of data increases. When the internal variability has a normal distribution, the spurious trend also has a normal distribution. In general cases of non-normal distributions, we obtain a formula of Edgeworth expansion for the distribution function of the spurious trend. In order to obtain the distribution function by the Edgeworth expansion, we need to know some moments of the internal variability.
To examine the atmospheric internal variabilities having a non-normal distribution, 15,200-year data are obtained by a numerical experiment with a simple global circulation model with purely periodic annual forcing. Dependence of estimation error of sample moments on the data length is computed to evaluate appropriate sample size for each moment. Spatial and seasonal variations of the moments of the internal variability are also described with the long dataset. There are some regions and seasons in which skewness and kurtosis is large and the variation is characterized by a non-normal distribution.
An ensemble experiment with the same model is done to argue the detectability of a cooling trend in the stratosphere from a finite-length dataset with internal variability. Standard deviation of the spurious trend is small in summer stratosphere and the estimated cooling trend is statistically significant because of the small internal variability.
Hypothesis tests for statistical significance of the estimated trend are also done; t-test and bootstrap test are compared with more accurate test with the distribution function derived by the Edgeworth expansion.
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