| Summary: | Forecasts of extreme events are useful in order to prepare for disaster. Such forecasts are usefully communicated as an upper quantile function, and in the presence of predictors, can be estimated using quantile regression techniques. This dissertation proposes methodology that seeks to produce forecasts that (1) are consistent in the sense that the quantile functions are valid (non-decreasing); (2) are flexible enough to capture the dependence between the predictors and the response; and (3) can reliably extrapolate into the tail of the upper quantile function. To address these goals, a family of proper scoring rules is first established that measure the goodness of upper quantile function forecasts. To build a model of the conditional quantile function, a method that uses pair-copula Bayesian networks or vine copulas is proposed. This model is fit using a new class of estimators called the composite nonlinear quantile regression (CNQR) family of estimators, which optimize the scores from the previous scoring rules. In addition, a new parametric copula family is introduced that allows for a non-constant conditional extreme value index, and another parametric family is introduced that reduces a heavy-tailed response to a light tail upon conditioning. Taken together, this work is able to produce forecasts satisfying the three goals. This means that the resulting forecasts of extremes are more reliable than other methods, because they more adequately capture the insight that predictors hold on extreme outcomes. This work is applied to forecasting extreme flows of the Bow River at Banff, Alberta, for flood preparation, but can be used to forecast extremes of any continuous response when predictors are present. === Science, Faculty of === Statistics, Department of === Graduate
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