Figure 3 - available via license: Creative Commons Attribution 4.0 International

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# Two perspectives of the same plot to illustrate issues associated with the lower bound imposed by the GEV fit: blue dots correspond to the top 70% NO 2 observations, used to fit the model in (5.1). Red squares (not used in the model fitting) are the remaining lower 30%. The grey surface correspond to the fitted lower boundˆµboundˆ boundˆµ(x)− ˆ σ(x)/ ˆ ξ(x) where x = (x 1 , x 2 ) and x 1 and x 2 are the vectors of monthly maximum temperature and wind speed, respectively. Five plausible combinations of covariate values and response (red squares) are not contained within the estimated support (i.e., above the fitted lower bound).

Source publication

The generalised extreme value (GEV) distribution is a three parameter family that describes the asymptotic behaviour of properly renormalised maxima of a sequence of independent and identically distributed random variables. If the shape parameter $\xi$ is zero, the GEV distribution has unbounded support, whereas if $\xi$ is positive, the limiting d...

## Contexts in source publication

**Context 1**

... n, where β µ,1 and β σ,1 are intercepts, β µ,2 is a linear regression coefficient and s µ,1 is a smooth function with four effective nonlinear degrees of freedom (see e.g., Hastie and Tibshirani, 1990). Figure 3 shows the data used to fit the model (blue dots) along with the fitted lower bound surfacê ...

**Context 2**

... these observations, the temperature ranges between 16.7 and 32.4 • C and the wind speed ranges between 4.2 and 16.2 m/s. We can see from Figure 3 that five of such observations fall below the fitted lower bound surface. In other words, a set of reasonable covariate and response values (red squares) is not contained within the estimated support, i.e., above the fitted lower bound (grey surface). ...

**Context 3**

... illustration shows the lack of robustness of the usual GEV distribution. Indeed, we fitted the GEV and bGEV distributions assuming that the only data we have are the blue dots in Figure 3. If we then observe new information (the red squares), then the GEV model is no longer a suitable representation of the distribution of maximum NO 2 concentrations, as there are at least four new observed values over which the GEV distribution assigns zero probability. ...