The Constrained Extremal Distribution Selection Method

Engineering design and policy formulation often involve the assessment of the likelihood of future events commonly expressed through a probability distribution. Determination of these distributions is based, when possible, on observational data. Unfortunately, these data are often incomplete, biased, and/or incorrect. These problems are exacerbated when policy formulation involves the risk of extreme events—situations of low likelihood and high consequences. Usually, observational data simply do not exist for such events. Therefore, determination of probabilities which characterize extreme events must utilize all available knowledge, be it subjective or observational, so as to most accurately reflect the likelihood of such events. Extending previous work on the statistics of extremes, the Constrained Extremal Distribution Selection Method is a methodology that assists in the selection of probability distributions that characterize the risk of extreme events using expert opinion to constrain the feasible values for parameters which explicitly define a distribution. An extremal distribution is then “fit” to observational data, conditional that the selection of parameters does not violate any constraints. Using a random search technique, genetic algorithms, parameters that minimize a measure of fit between a hypothesized distribution and observational data are estimated. The Constrained Extremal Distribution Selection Method is applied to a real world policy problem faced by the U.S. Environmental Protection Agency. Selected distributions characterize the likelihood of extreme, fatal hazardous material accidents in the United States. These distributions are used to characterize the risk of large scale accidents with numerous fatalities.