Small-sample likelihood inference in extreme-value regression models

Abstract

We deal with a general class of extreme-value regression models introduced by Barreto-Souza and Vasconcellos [Bias and skewness in a general extreme-value regression model, Comput. Statist. Data Anal. 55 (2011), pp. 1379–1393]. Our goal is to derive an adjusted likelihood ratio statistic that is approximately distributed as χ² with a high degree of accuracy. Although the adjusted statistic requires more computational effort than its unadjusted counterpart, it is shown that the adjustment term has a simple compact form that can be easily implemented in standard statistical software. Further, we compare the finite-sample performance of the three classical tests (likelihood ratio, Wald, and score), the gradient test that has been recently proposed by Terrell [The gradient statistic, Comput. Sci. Stat. 34 (2002), pp. 206–215], and the adjusted likelihood ratio test obtained in this article. Our simulations favour the latter. Applications of our results are presented.

Keywords:

• extreme-value regression
• gradient test
• Gumbel distribution
• likelihood ratio test
• nonlinear models
• score test
• small-sample adjustments
• Wald test

Acknowledgements

We thank the referee for helpful comments and suggestions. We gratefully acknowledge the financial support from CNPq.

Notes

1. For linear extreme-value regression models with constant dispersion, the null distributions of the five statistics do not depend on φ. The proof is omitted to save space.

2. The data set is also available at http://www.stat.ncsu.edu/working_groups/sas/sicl/data/olympic.dat.