| Abstract: | Tail estimators are proposed which make minimal assumptions and let the data dictate the form of the probability model. These estimators use only the observations in the tail and are based on a unifying density-quantile model. The fundamental result in this work is a representation of the quantile function of the exceedences over a threshold. This representation: (1) motivates a unified parameterization for tail estimators of the underlying probability model; (2) motivates methods for obtaining parameter estimates; and (3) simplifies the derivation of the asymptotic properties of the proposed parameter estimates. Parameter estimates may be obtained using a Generalized Pareto Distribution or a Generalized Extreme Value Distribution model of the exceedences. Assuming the underlying distribution can be correctly classified as either short tailed or long tailed, other estimates are formed. The asymptotic properties of these estimates are derived under rate of convergence conditions to show the effect of threshold selection on parameter properties. The parameters are shown to be nonidentifiable and their estimators contain a bias which may approach zero very slowly. Therefore, if the parameters are the focus of the analysis, extremely large sample sizes are required to reduce the bias to a negligible amount. If the tail estimates are of interest, the bias is less likely to be serious and the nonidentifiability problem provides a closer approximation to the tail for small samples. |
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