Descriptive statistics for parametric models currently rely heavily on the accuracy of distributional assumptions. Here, leveraging the structures of parametric distributions and their central moment kernel distributions, a class of estimators, consistent simultanously for both a semiparametric distribution and a distinct parametric distribution, is proposed. These efficient estimators are robust to both gross errors and departures from parametric assumptions, making them ideal for estimating the mean and central moments of common unimodal distributions. This article also illuminates the understanding of the common nature of probability distributions and the measures of them.

These papers are prepared for PNAS. The related codes and drafts were shared and have been publically posted on my Github one year ago. I am introducing this work in YouTube and Quora, if you are interested, please visit: https://www.youtube.com/@Iobiomathematics or https://www.quora.com/profile/Tuobang-Li-1/answers or https://www.researchgate.net/profile/Tuobang-Li-2. For more information, please visit https//github.com/tubanlee. Also, feel free to share it or contact tl@biomathematics.org, for more materials available by request.