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

Abstract

Suppose that a statistician observes two independent variates X1X_1 and X2X_2 having densities fi(;θ)fi(θ) , i=1,2f_i(\cdot;\theta)\equiv f_i(\cdot-\theta)\ ,\ i=1,2 , θR\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 α(0,1)\alpha\in(0,1). Moran (1973) suggested a test which is based on a single split of the data, \textit{i.e.,} to use X2X_2 in order to conduct a one-sided test in the direction of X1X_1. Specifically, if b1b_1 and b2b_2 are the (1α)(1-\alpha)'th and α\alpha'th quantiles associated with the distribution of X2X_2 under HH, 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 aRa\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 f1()f_1(\cdot) and f2()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 dd-dimensional Gaussians.

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