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Nor Maximum Likelihood Estimators (MLE) neither the Moments Matching Estimators (MME) for the two parameters $\alpha, \beta$ (shape and rate respectively) are unbiased.

Is there a closed formula to compute an unbiased estimator for both the shape and rate of the gamma distribution?

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No closed form for an unbiased estimator is known. The most recent work on closed form bias correction (which still leaves some bias in small samples) seems to be Louzada et al. (2019, The American Statistician).

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