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

Suppose that a statistician observes two independent variates X₁ and X₂ having densities f_i (·; θ) ≡ f_i (·−θ),i=1, 2, θ ∈ R. His purpose is to conduct a test for H: θ =0 vs. K: θ ∈ R \{0} with a pre-defined significance level α ∈ (0, 1). Moran (1973) suggested a test which is based on a single split of the data, i.e., to use X₂ in order to conduct a one-sided test in the direction of X₁. Specifically, if b₁ and b₂ are the (1 − α)’th and α’th quantiles associated with the distribution of X₂ under H, then Moran’s test has a rejection zone (a, ∞) × (b₁, ∞) ∪(−∞,a) × (−∞,b₂) where a ∈ 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 f₁(·) andf₂(·) under which Moran’s test is inadmissible.