On higher-order moment and cumulant estimation

Moments and cumulants are involved in statistical analysis for a wide range of fields. A natural and popular approach to moment and cumulant estimation is based on the sample average. However, it is well known that these sample estimates usually perform poorly. In this paper, we derive uniformly minimum-variance unbiased estimator for raw moment, centred moment, and cumulant of any order for a number of common distributions. Extensive simulation studies demonstrate that the proposed estimators can perform much better than the corresponding sample average estimators.