TABLE D-2 Similar to Table D-1, but for the Russia-Pakistan Dataset

Logistic Model Ramos-Ledford Model

Estimate 90% CI Estimate 90% CI

10-year 1.01 (1.00, 1.01) 0.33 (0.04, 1.4)
20-year 1.02 (1.00, 1.03) 0.21 (0.008, 1.8)
50-year 1.05 (1.01, 1.07) 0.17 (0.001, 2.9)

NOTE: Shown are the point estimate and 90 percent confidence interval, under both the logistic model and the Ramos-Ledford model.

In contrast with Figure D-1, the right hand plot of Figure D-2 shows virtually no data point away from the axes, indicating that there is no evidence of dependence in the upper tail of the distribution. This is confirmed by repeating the same analyses as for Example 1, with results shown in Table D-2. For the logistic model, which is constrained to positive dependence between the two variables, the point estimates and confidence intervals (for the ratio of joint probability to the independent case) are all very close to 1. Under the Ramos-Ledford model, which does not have that constraint, the estimated probability ratios are <1 (indicating negative dependence), but the confidence intervals include 1. With either set of results, the net conclusion is that there is no evidence against the hypothesis of independence in the right hand tail of the distribution.

Conclusions. Example 1 confirms and extends the results of Herweijer and Seager (2008) by showing that the interdependence of drought conditions in the two given regions of the United States and South America extends to the tail of the distribution, although the confidence intervals for the probability ratios are still fairly wide as a result of the relatively small number of data points (102). However, Example 2 shows no evidence at all that there is any tendency for extreme high temperatures in Russia to be associated with extreme high precipitation in Pakistan; in other words, the 2010 event may have been truly an outlier without precedent in history. This should however be qualified by noting that the dataset used, consisting of monthly averages over half-degree grid cells, cannot be expected to reproduce extreme precipitation events over very short time and spatial scales, and it remains possible that an alternative data source, using finer-scale data, would produce a different conclusion.


There is a substantial body of statistical literature on univariate and bivariate extremes and more limited research on extremes in higher dimensions. However, practical application of these methods in extreme value

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