Skewness and Kurtosis for Maximum Likelihood Estimator in one-parameter Exponential Family Models

In this paper we derive approximate formulae for the skewness and kurtosis of the maximum likelihood estimator in the one-parameter exponential family. The key idea underlying these formulae is that they indicate when the normal approximation usually employed with maximum likelihood estimators can be misleading in small samples. We apply our main result to a number of special distributions of this family. We also use a graphical analysis to examine how the skewness and kurtosis vary with the true value of the parameter in some special cases.

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