Bias reduction in one-parameter exponential family models

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This paper gives closed-form expressions for the second and third order biases of maximum likelihood estimates for a number of distributions in the one-parameter exponential family. Approximations based on asymptotic expansions for some bias-corrections that require the evaluation of unusual functions and long expressions are given. A graphical analysis is also performed to show how such biases and the mean squared errors of both the maximum likelihood estimate and its bias-corrected version vary with the parameter that indexes some distributions. Finally, we present simulation results comparing the performance of the maximum likelihood estimator and its bias-corrected version.