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be an iid sequence of random variables again with

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Unformatted text preview: he # fitted object, ’fit,’ in order to use the "extRemes" # function, ’return.level’. class( fit) < − "gev.fit" fit.rl < − return.level( fit) Return Estimated Return 95% CI Period Level (in) 10 2.81 (2.41, 3.21) 100 5.10 ?(3.35, 6.84) . . . . . . Example Fort Collins, Colorado precipitation Return Levels CI’s from return.level are based on the delta method, which assumes normality for the return levels. For longer return periods (e.g., beyond the range of the data), this assumption may not be valid. Can check by looking at the profile likelihood. gev.prof( fit, m=100, xlow=2, xup=8) Highly skewed! Using locator(2), a better approximation for the (95%) 100-year return level CI is about (3.9, 8.0). Example Fort Collins, Colorado precipitation Probability of annual maximum precipitation at least as large as that during the 28 July 1997 flood (i.e., Pr{max(X ) ≥ 1.54 in.}). # Using the ’pgev’ function from the "evd" package. pgev( 1.54, loc=fit$mle[1], scale=fit$mle[2], shape=fit$mle[3], lower.tail=FALSE) pgev( 4.6, loc=fit$mle[1], scale=fit$mle[2], shape=fit$mle[3], lower.tail=FALSE) Peaks Over Thresholds (POT) Approach Let X1, X2, . . . be an iid sequence of random variables, again with marginal distribution, F . Interest is now in the conditional probability of X ’s exceeding a certain value, given that X already exceeds a sufficiently large threshold, u. Pr{X > u + y |X > u} = 1 − F (u + y ) ,y>0 1 − F (u ) Once again, if we know F , then the above probability can be computed. Generally not the case in practice, so we turn to a broadly applicable approximation. Peaks Over Thresholds (POT) Approach If Pr{max{X1, . . . , Xn} ≤ z } ≈ G(z ), where G(z ) = exp − 1 + ξ z−µ σ −1/ξ for some µ, ξ and σ > 0, then for sufficiently large u, the distribution [X − u|X > u], is approximately the generalized Pareto distribution (GPD). Namely, ξy H (y ) = 1 − 1 + ˜ σ −1/ξ , y > 0, + ˜ with σ = σ + ξ (u − µ) (σ , ξ and µ as in G(z ) above). Peaks Over Thresholds (POT) Approach GPD 0.1 0.2 0.3 0.4 0.5 Pareto (xi>0) Beta (xi<0) Exponential (xi=0) 0.0 • Exponential type (ξ = 0) light tail pdf • Beta type (ξ < 0) bounded above at u − σ/ξ 0.6 • Pareto type (ξ > 0) heavy tail 0 2 4 6 8 Peaks Over Thresholds (POT) Approach Hurricane damage Economic Damage from Hurricanes (1925−1995) 40 Trends in societal vulnerability removed. 20 q q q q qq 0 billion US$ 60 q Economic damage caused by hurricanes from 1926 to 1995. q q q q q q q qq q q q q q q qqq q qq qq q q q q q qq q qq q qq q q qqq qqqqqqqqqqqqqq qqqqqqqq q qqqq qqqqqqq qqqqqqq qqq qq qq qq q q qq q q q 1930 1940 1950 1960 Year 1970 1980 1990 Excess over threshold of u = 6 billion US$. Peaks Over Thresholds (POT) Approach Hurricane damage GPD 0.20 Likelihood ratio test for ˆ σ =≈ 4.589 ˆ ξ ≈ 0.512 0.05 0.10 95% CI for shape parameter using profile likelihood. 0.05 < ξ < 1.56 0.00 pdf 0.15 ξ = 0 (p-value≈ 0.018) 10 20 30 40 50 60 70 Peaks Over Thresholds (POT) Approach Choosing a threshold Variance/bias trade-off Low threshold allows for more data (low variance). 10 q q q 5 q q q q q 0 q q −5 Modified Scale Theoretical justification for GPD requires a high threshold (low bias). 2 3 4 5 6 7 0.5 q q q q q q q q q q 0.0 Shape 1.0 Threshold 2 3 4 5 Threshold 6 7 gpd.fitrange( damage$Dam, 2, 7) Peaks Over Thresholds (POT) Approach Dependence above threshold Often, threshold excesses are not independent. For example, a hot day is likely to be followed...
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