In 1954, Hodges and Lehmann demonstrated that if X and Y are independently sampled from an identical unimodal distribution, X−Y will exhibit symmetrical unimodality with its peak centered at zero. Building upon this foundational work, the current study delves into the structure of the kernel distribution of U -statistics. It is shown that the kth central moment kernel distributions (k > 2) derived from a unimodal distribution exhibit location invariance and is also nearly unimodal with the mode and median close to zero. This article provides an approach to study the general structure of kernel distributions.
These works have been publically deposited in this Github since one year ago for a PNAS paper (I hidden some previous versions after updated new versions, https://github.com/tubanlee/NRS3333). 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 . Also, the manuscript has been deposited in Zenodo Tuobang Li. (2022). Robust estimations for semiparametric models: Moments. https://doi.org/10.5281/zenodo.8127703 and research gate https://www.researchgate.net/publication/377974264_Robust_estimations_from_distribution_structures_II_Central_Moments
Feel free to share it or contact tl@biomathematics.org, for more materials available by request.
This is originally combined with REDS: Invariant Moments into a single paper, but now I think it is better to present it as a single paper.
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