Optimal unbiased estimation of some population central moments

Abstract

In this paper, we have treated the problem of estimating some population central moments under distribution-free setting. Uniformly minimum variance unbiased estimators for some population central moments have been derived. Some examples of unbiased estimators of central moments have been given under various sampling designs such as simple random sampling with replacement (srsr) or without replacement (srs), probability proportional to size with replacement (ppsr) and probability graduated variable proportional to size without replacement (pgvps). An optimal unbiased estimator of the third population central moment is proposed and extended to some real situations. Some optimal unbiased estimators of the fourth population central moment are given. Several optimal unbiased estimators of the variance of the “sample quasivariance estimator” are identified. Finally, computer programs in R implementing all of the estimators are given.