Nonparametric Analysis of Temporal Trend When Fitting Parametric Models to Extreme­Value Data

Abstract

A topic of major current interest in extreme­value analysis is the investigation of temporal trends. For example, the potential influence of “greenhouse” effects may result in severe storms becoming gradually more frequent, or in maximum temperatures gradually increasing, with time. One approach to evaluating these possibilities is to fit, to data, a parametric model for temporal parameter variation, as well as a model describing the marginal distribution of data at any given point in time. However, structural trend models can be difficult to formulate in many circumstances, owing to the complex way in which different factors combine to influence data in the form of extremes. Moreover, it is not advisable to fit trend models without empirical evidence of their suitability. In this paper, motivated by datasets on windstorm severity and maximum temperature, we suggest a nonparametric approach to estimating temporal trends when fitting parametric models to extreme values from a weakly dependent time series. We illustrate the method through applications to time series where the marginal distributions are approximately-Pareto, generalized­Pareto, extreme­value or Gaussian. We introduce time­varying probability plots to assess goodness of fit, we discuss local­likelihood approaches to fitting the marginal model within a window and we propose temporal cross­validation for selecting window width. In cases where both location and scale are estimated together, the Gaussian distribution is shown to have special features that permit it to playa universal role as a “nominal” model for the marginal distribution.