Exercise Solution 14.5

Historical exceedance data ^– αi, ^{– α +1}i, … , ⁰i comprises a sequence of 0s and 1s. Because 0s are more numerous than 1s, the sequence usually starts and ends with 0. In that case, every pair of consecutive observations (^t–1i, ^ti) of form (0, 1) must be followed by another of form (1, 0). The latter may occur immediately, if there are three consecutive data points 0, 1, 0. It may be delayed if several 1s occur consecutively, as in 0, 1, 1, 1, 0. But because every pair of consecutive observations of form (0, 1) is eventually followed by another of form (1, 0), the total α₀₁ of those having form (0, 1) must equal the total α₁₀ of those having form (1, 0).

Three less common situations are that historical exceedance data ^– αi, ^{– α +1}i, … , ⁰i

• starts with a 0 and ends with a 1, in which case α₀₁ = α₁₀ + 1;
• starts with a 1 and ends with a 0, in which case α₀₁ = α₁₀ – 1;
• starts with a 1 and ends with a 1, in which case α₀₁ = α₁₀.