  In descriptive statistics, $U$-statistics arise naturally in producing
minimum-variance unbiased estimators. In 1984, Serfling considered the
distribution formed by evaluating the kernel of the $U$-statistics and proposed
generalized $L$-statistics which includes Hodges-Lehamnn estimator and
Bickel-Lehmann spread as special cases. However, the structures of the kernel
distributions remain unclear. In 1954, Hodges and Lehmann demonstrated that if
$X$ and $Y$ are independently sampled from the same 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. It is shown that the $\mathbf{k}$th
central moment kernel distributions ($\mathbf{k}>2$) 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.
