In this Supplementary Material, we have two sections. The first part gives the detailed derivations of the UMVUEs for the central moments, rawo moments and the cumulants when data are generated from the (negative) Binomial distribution, the Poisson distribution, the discrete/continuous uniform distribution and the exponential distribution. The second part provides extensive simulation studies to compare the performances of our proposed estimators with the maximum likelihood estimators (MLEs), the sample average estimators and the h/k-statistics.
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