Median bias reduction of maximum likelihood estimates

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.