Appendix E. Estimating the shape of tails.
The shape of the tail of maximal standardized residuals can be estimated from the data in hand. For example, Fig. E1 shows the cumulative distribution of the largest 0.2% of standardized residuals for predicted wave periods, a distribution of “points above threshold” that, in theory, asymptotically approaches a generalized Pareto model (Coles 2001):
Here x is the variable of interest (in this example the standardized wave period residuals), u is the threshold value of x, and K and M are fitted shape and scale parameters, respectively. If the best-fit estimate of K is > 0, the distribution has an unbounded tail in which exceptionally large values may hide. If K = 0, the tail is unbounded, but with few exceptional values. If K < 0, the tail has a distinct upper bound. In the case of wave period standardized residuals, K is significantly less than 0 (-0.313 ± 0.054 [95% CL]), indicating that the tail of the distribution is bounded. A similar result obtains for the normalized deviation of tidal height residuals (K = -0.255 ± 0.072 [95% CL]). The best-fit estimate of K for significant wave height is negative, but not significantly different from 0 (K = -0.041 ± 0.045 [95% CL]). In summary, our short-term estimates for the distribution of standardized tide and wave residuals undoubtedly miss some of the large residuals that would be captured by longer-term records, but it appears unlikely that exceptionally large residuals are hidden in the unsampled tails of these distributions.
The calculation of limpet body temperature involves additional environmental variables. The tails for the standardized residuals of air temperature and sea-surface temperature are unbounded (K = 0.027 ± 0.009, 0.329 ± 0.006, respectively). The exceptional values found in the unsampled portions of these distributions might contribute to higher body temperatures than we have calculated here.
Coles, S. T. 2001. An introduction to statistical modeling of extreme values. Springer-Verlag. London, UK.
|FIG E1. The upper tail of wave period deviations closely matches a generalized Pareto function (Eq. E1).|