  In numerous regular statistical models, median bias reduction (Kenne Pagui et
al., 2017) has proven to be a noteworthy improvement over maximum likelihood,
alternative to mean bias reduction. The estimator is obtained as solution to a
modified score equation ensuring smaller asymptotic median bias than the
maximum likelihood estimator. This paper provides a simplified algebraic form
of the adjustment term for general regular models. With the new formula, the
estimation procedure benefits from a considerable computational gain by
avoiding multiple summations and thus allows an efficient implementation. More
importantly, the new formulation allows to highlight how the median bias
reduction adjustment can be obtained by adding an extra term to the mean bias
reduction adjustment. Illustrations are provided through new applications of
median bias reduction to two regression models not belonging to the generalized
linear models class, extended beta regression and beta-binomial regression.
Mean bias reduction is also provided here for the latter model. Simulation
studies show remarkable componentwise median centering of the median bias
reduced estimator, while variability and coverage of related confidence
intervals are comparable with those of mean bias reduction. Moreover, empirical
results for the beta-binomial model show that the method is successful in
solving maximum likelihood boundary estimate problem.
