Determining exceedance regions, such as regions where a specified threshold of a pollutant in the environment is exceeded, is of critical importance for decision-making in environmental management and public health. Inner and outer predicted exceedance sets express the uncertainties in predicted exceedance regions as they sandwich the unknown true exceedance region with high confidence, analogous to confidence regions for point estimates. It is therefore desirable to reduce the uncertainty about the locations of the true exceedance region, resulting in a narrow band between the inner and outer sets. However, in practice this is not often the case mainly due to the strict statistical subset criteria being set, which are equivalent to a multiple testing problem controlling the familywise error rate (FWER). It is well known that the FWER leads to fewer rejections compared to other criteria; in the context of exceedance regions, this would correspond to an extremely small, conservative inner predicted exceedance region. In this paper, we loosen the criteria slightly to obtain a narrower band between inner and outer sets, allowing for more nuanced uncertainty assessments. A new algorithm is proposed to construct these exceedance sets, and the methods are compared in a simulation study to assess whether they indeed control the new criteria. The methods are illustrated on two data sets: average rainfall in the state of Paraná, Brazil, and nitrogen dioxide air pollution in Germany in the year 2018.
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