  For regular parametric problems, we show how median centering of the maximum
likelihood estimate can be achieved by a simple modification of the score
equation. For a scalar parameter of interest, the estimator is equivariant
under interest respecting parameterizations and third-order median unbiased.
With a vector parameter of interest, componentwise equivariance and third-order
median centering are obtained. Like Firth's (1993, Biometrika) implicit method
for bias reduction, the new method does not require finiteness of the maximum
likelihood estimate and is effective in preventing infinite estimates.
Simulation results for continuous and discrete models, including binary and
beta regression, confirm that the method succeeds in achieving componentwise
median centering and in solving the infinite estimate problem, while keeping
comparable dispersion and the same approximate distribution as its main
competitors.
