  Suppose that a statistician observes two independent variates $X_1$ and $X_2$
having densities $f_i(\cdot;\theta)\equiv f_i(\cdot-\theta)\ ,\ i=1,2$ ,
$\theta\in\mathbb{R}$. His purpose is to conduct a test for
  \begin{equation*}
  H:\theta=0 \ \ \text{vs.}\ \ K:\theta\in\mathbb{R}\setminus\{0\}
  \end{equation*}
  with a pre-defined significance level $\alpha\in(0,1)$.
  Moran (1973) suggested a test which is based on a single split of the data,
\textit{i.e.,} to use $X_2$ in order to conduct a one-sided test in the
direction of $X_1$. Specifically, if $b_1$ and $b_2$ are the $(1-\alpha)$'th
and $\alpha$'th quantiles associated with the distribution of $X_2$ under $H$,
then Moran's test has a rejection zone
  \begin{equation*}
  (a,\infty)\times(b_1,\infty)\cup(-\infty,a)\times(-\infty,b_2)
  \end{equation*}
  where $a\in\mathbb{R}$ is a design parameter.
  Motivated by this issue, the current work includes an analysis of a new
notion, \textit{regular admissibility} of tests. It turns out that the theory
regarding this kind of admissibility leads to a simple sufficient condition on
$f_1(\cdot)$ and $f_2(\cdot)$ under which Moran's test is inadmissible.
Furthermore, the same approach leads to a formal proof for the conjecture of
DiCiccio (2018) addressing that the multi-dimensional version of Moran's test
is inadmissible when the observations are $d$-dimensional Gaussians.
