Simple sufficient condition for inadmissibility of Moran's single-split test
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Suppose that a statistician observes two independent variates X_1 and X_2 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 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-α)'th and α'th quantiles associated with the distribution of X_2 under H, then Moran's test has a rejection zone (a,∞)×(b_1,∞)∪(-∞,a)×(-∞,b_2) 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 f_1(·) and f_2(·) 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.
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