In the fields of structural safety and structural dynamics, there are many areas in which it is often desirable to compute the first few statistical moments of a performance function. In previous studies, five- and seven-point estimates for statistical moments have been proposed, in which the estimating points are independent of the random variable in original space and high-order moments of the original random variables are not required. These methods have been verified to be generally efficient for estimating the first two moments of a function of random variables. In the present study, in order to improve the estimation accuracy for high-order moments, the general relationship between the weights and concentrations in arbitary number are derived, and the concentrations up to 21 points for a function of a single random variable as well as the formulas for the first six moments of a function of multiple random variables are presented. From several numerical examples, it is shown that high-order moments can be estimated with high accuracy using the proposed formulations.