distributed, but with changing means, they provided concrete formulas for calculating the probability that a specific threshold is exceeded over a given period of time. However, the restriction to independent observations limits the usefulness of their approach.

An alternative is to use Monte Carlo simulation. Based on any fitted time series model, such as an autoregressive moving average (ARMA) model with time-varying means, one could calculate future extreme event probabilities by simulating the time series many times and calculating the proportion of simulations for which the extreme event of interest occurs. The principal limitation of such methods lies not in the simulation itself, which is fast and accurate using modern computing techniques, but in the structure of the time series model; if this is misspecified, then the extreme value probabilities may be over- or under-estimated by orders of magnitude.

In particular, we question whether Gaussian probabilities are appropriate for extreme events. In the case of temperature series, simple plots of the data do show an approximately normal shape (Hansen et al., 2012, have several examples), but this does not preclude the possibility of some extreme events that are caused by natural variation in the weather. An example is given by Dole et al. (2011), where they dispute the assertion that the 2010 Russian heat wave was associated with anthropogenic climate change. A key point of their argument was the presence of a blocking event, which could be a natural occurrence, but one that is expected to lead to much more extreme temperatures than usual. In the presence of such phenomena, one would not expect the distributions of the most extreme events to be consistent with Gaussian probabilities. For other kinds of meteorological variables, such as precipitation and wind speed, it is not realistic to assume Gaussian probabilities at all.

METHODS BASED ON EXTREME VALUE THEORY

An alternative class of statistical methods, which do not assume Gaussian probabilities, is the class of methods based on the statistical theory of extreme values. One very readable book on this topic is by Coles (2001). According to classical extreme value theory, the distribution of maxima over a fixed time period (e.g., annual maximum temperatures) may be approximated by a family of probability distributions known as extreme value distributions. These may be summarized in the form of the Generalized Extreme Value (GEV) distribution, which is characterized by three parameters: a location parameter representing the center of the distribution, a scale parameter representing variability, and a third “shape” parameter, which is the key parameter in characterizing probabilities of very extreme events. Although in their original form such methods were developed for stationary (but not necessarily Gaussian) time series, they



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