Adjusted profile likelihoods for the weibull shape parameter

Abstract

This paper presents several different adjusted profile likelihoods for the Weibull shape parameter. These adjustments aim at reducing the impact of the nuisance parameter on the likelihood-based inference regarding the parameter of interest. Both point estimation and hypothesis testing are considered. We also show that the ratio between the estimators and the shape parameter are pivotal quantities and that the size properties of the usual and adjusted profile likelihood ratio tests depend neither on the scale parameter nor on the value of the shape parameter set at the null hypothesis. The numerical results suggest that the adjustment obtained by Yang and Xie [Yang, Z. and Xie, M., 2003, Efficient estimation of the Weibull shape parameter based on a modified profile likelihood. Journal of Statistical Computation and Simulation, 73, 115–123.] outperforms not only the profile likelihood inference but also inference based on competing adjusted profile likelihoods.

Keywords:

• Censoring
• Failure time data
• Likelihood ratio test
• Maximum likelihood estimator
• Nuisance parameter
• Profile likelihood
• Weibull distribution

Acknowledgements

We gratefully acknowledge the partial financial support from CNPq and FAPESP. We also thank an anonymous referee for comments on an earlier draft.

Notes

^†Throughout this section α and β are allowed to be vector-valued. However, in the model of interest in section 4 these paramaters are scalars.