Simple sufficient condition for inadmissibility of Moran’s single-split test

Abstract

Suppose that a statistician observes two independent variates $X_1$ and $X_2$ having densities $f_i(\cdot ;\mathit{\theta })\equiv f_i(\cdot -\mathit{\theta }),i=1,2$, $\mathit{\theta }\in \mathbb{R}$. His purpose is to conduct a test for

$$H:\mathit{\theta }=0\text{vs.}K:\mathit{\theta }\in \mathbb{R}\setminus \{0\}$$

with a pre-defined significance level $\mathit{\alpha }\in (0,1)$. Moran (1973) suggested a test which is based on a single split of the data, 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 $(1-\mathit{\alpha })$’th and α’th quantiles associated with the distribution of $X_2$ under H, then Moran’s test has a rejection zone

$$(a,\mathrm{\infty })\times (b_1,\mathrm{\infty })\cup (-\mathrm{\infty },a)\times (-\mathrm{\infty },b_2)$$

where $a\in \mathbb{R}$ is a design parameter. Motivated by this issue, the current work includes an analysis of a new notion, 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.