Suppose that a statistician observes two independent variates and having densities , . 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 . Moran (1973) suggested a test which is based on a single split of the data, \textit{i.e.,} to use in order to conduct a one-sided test in the direction of . Specifically, if and are the 'th and 'th quantiles associated with the distribution of under , 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 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 and 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 -dimensional Gaussians.
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