Selection of the k Largest Order Statistics for the Domain of Attraction of the Gumbel Distribution

Abstract

Statistical modeling of the largest k observations or the exceedances over a high threshold based on extreme value theory has produced powerful methods for drawing inferences about the tail behavior of a distribution; however, a method for selecting k has not been established. We suggest using a forward selection procedure based on a suitable goodness-of-fit statistic to search for the optimal k when the sample is of finite size and from a distribution in the domain of attraction of the Gumbel distribution. Two criteria are examined: generalized least squares and the Shapiro-Wilk goodness of fit. Simulation studies indicate that the latter is preferable in terms of variation. Extension of the selection procedure to distributions in the domains of attraction of the Fréchet and Weibull distributions is also discussed. The use of the proposed method is illustrated through two examples based on real data.

Key Words:

• Coefficient of determination
• Extreme value
• k largest observations
• Shapiro-Wilk statistic