Objective Priors for Invariant e-Values in the Presence of Nuisance Parameters.

This paper aims to contribute to refining the e-values for testing precise hypotheses, especially when dealing with nuisance parameters, leveraging the effectiveness of asymptotic expansions of the posterior. The proposed approach offers the advantage of bypassing the need for elicitation of priors and reference functions for the nuisance parameters and the multidimensional integration step. For this purpose, starting from a Laplace approximation, a posterior distribution for the parameter of interest is only considered and then a suitable objective matching prior is introduced, ensuring that the posterior mode aligns with an equivariant frequentist estimator. Consequently, both Highest Probability Density credible sets and the e-value remain invariant. Some targeted and challenging examples are discussed.

Improved estimation in negative binomial regression.

Negative binomial regression is commonly employed to analyze overdispersed count data. With small to moderate sample sizes, the maximum likelihood estimator of the dispersion parameter may be subject to a significant bias, that in turn affects inference on mean parameters. This article proposes inference for negative binomial regression based on adjustments of the score function aimed at mean or median bias reduction. The resulting estimating equations generalize those available for improved inference in generalized linear models and can be solved using a suitable extension of iterative weighted least squares. Simulation studies confirm the good properties of the new methods, which are also found to solve in many cases numerical problems of maximum likelihood estimation. The methods are illustrated and evaluated using two case studies: an Ames salmonella assay data set and data on epileptic seizures. Inference based on adjusted scores turns out to generally improve on maximum likelihood, and even on explicit bias correction, with median bias reduction being overall preferable.

Adjusted score functions for monotone likelihood in the Cox regression model.

Standard inference procedures for the Cox model involve maximizing the partial likelihood function. Monotone partial likelihood is an issue that frequently happens in the analysis of health science studies. Monotone likelihood mainly occurs in samples with substantial censoring of survival times and is associated with categorical covariates. In particular, and more frequently, it usually happens when one level of a categorical covariate has just experienced censoring times. In order to overcome this problem, Heinze and Schemper proposed an adjusted partial likelihood score function obtained by suitably adapting the general approach of Firth for mean bias reduction. The procedure is effective in preventing infinite estimates. As an alternative solution, we propose an approach based on the adjusted score function recently suggested by Kenne Pagui et al for median bias reduction. This procedure also solves the infinite estimate problem and has an additional advantage of being invariant under componentwise reparameterizations. This latter fact is fundamental under Cox model since hazards ratio interpretation is obtained by exponentiating parameter estimates. Numerical studies of the proposed method suggest better inference properties than those of the mean bias reduction. A real-data application related to a melanoma skin dataset is used as illustration for a comparison basis of the methods.