Adjusted profile likelihoods for the weibull shape parameter

Abstract

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 (2003) outperforms not only the profile likelihood inference but also inference based on competing adjusted profile likelihoods.

Improved likelihood inference for the shape parameter in Weibull regression

“…It is also shown that the size properties of the usual and adjusted profile likelihood ratio tests depend neither on φ nor on the value of σ set at the null hypothesis. We generalize some of the results in Ferrari et al (2005) and Yang and Xie (2003); unlike their results, ours allow for regression dependence. We also report numerical evidence on the finite-sample performance of the different inference strategies; it favors the inference obtained following Fraser and Reid (1995), Fraser et al (1999).…”

“…These parameterizations were obtained under noncensoring, thus it is not possible to guarantee parameter orthogonality when observations were recorded under either censoring scheme. The numerical evidences in Yang and Xie (2003) and in Ferrari et al (2005) on maximum likelihood estimation and likelihood ratio tests based on CR (β) showed that the resulting inference was reliable. In order to evaluate the effectiveness of Cox and Reid's (1987) adjustment to the Weibull profile likelihood function in the regression case, we considered parameterizations that are similar in nature to those of Yang and Xie (2003).…”

“…However, in small samples the resulting estimators can be considerably biased and likelihood ratio tests can be substantially size distorted. Profile likelihood adjustments for the Weibull shape parameter with no covariates were obtained by Ferrari et al (2005) and Yang and Xie (2003). Their Monte Carlo results suggest that adjusted profile maximum likelihood estimators are less biased and more efficient than the usual maximum likelihood estimator in small samples both with and without censoring.…”

“…It is noteworthy that, unlike Ferrari et al (2005) and Yang and Xie (2003), we consider the case of vector-valued nuisance parameter.…”

Transmutation maps: modeling, structural properties, estimation and applications

“…In our work, we particularly consider the presence of right censorship and, in order to t this new model, we used prole methods, such as pure, adjusted and modied prole; see, for example, BarndorNielsen and McCullagh (1993), Severini (1998) and Ferrari et al (2007). Note that the pure likelihood method cannot estimate the parameters in samples with high percentages of censored points, which is what our application case showed.…”