Asymptotic cumulants of the estimator of the canonical parameter in the exponential family

• Asymptotic cumulants the MLE for the canonical parameter are obtained.
• The skewness and kurtosis are used for interval estimation.
• The higher-order variances of the Jeffreys estimator and the MLE are compared.
• The mean square errors of the Jeffreys estimator and the MLE are compared.
• The kurtosis to squared skewness ratio is crucial to the mean square error.

Asymptotic cumulants of the maximum likelihood estimator of the canonical parameter in the exponential family are obtained up to the fourth order with the added higher-order asymptotic variance. In the case of a scalar parameter, the corresponding results with and without studentization are given. These results are also obtained for the estimators by the weighted score, especially for those using the Jeffreys prior. The asymptotic cumulants are used for reducing bias and mean square error to improve a point estimator and for interval estimation to have higher-order accuracy. It is shown that the kurtosis to squared skewness ratio of the sufficient statistic plays a fundamental role.