Linear Regression with Extreme Value Residuals

Abstract

We propose a method of estimating the p + 3 parameters of a random sample consisting of a trend with p covariates plus residuals from the three-parameter generalized extreme value distribution (GEV). The estimates are the p maximum likelihood estimates for the p-variate slope β, combined iteratively with the L-moment estimates for the three GEV parameters. An application is illustrated to annual maximum rainfall data from Orlando, Florida.

With p = 1, we show how to compute by simulation the densities and percentiles of and , where α is the GEV scale parameter and is its estimate. We do this for both ‘2-sample’ covariates and linear covariates, and for a range of values of the GEV shape parameter k. These two distributions are invariant to the location and scale parameters but do depend on the covariates. Also, their percentiles do not vary greatly for the shape parameter k near 0. Thus, the percentiles of provide a convenient confidence region for β at least when the shape parameter estimate is near 0. The method can be applied to any choice of covariates. A similar analysis is carried out for the Gumbel (EV1) distribution with trend, that is, for the case k = 0. A variation of this confidence region is suggested when p > 1.

Keywords:

• Extreme values
• GEV
• Percentiles
• Slope test
• Trend

Mathematics Subject Classification:

• 62J05

Acknowledgments

The authors would like to thank the Editor-in-Chief and the referee for their careful reading and comments which helped to greatly improved the article.