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

20 March 2021
Royi Jacobovic
ArXiv (abs)PDFHTML
Abstract

Suppose that a statistician observes two independent variates X1X_1X1​ and X2X_2X2​ having densities fi(⋅;θ)≡fi(⋅−θ) , i=1,2f_i(\cdot;\theta)\equiv f_i(\cdot-\theta)\ ,\ i=1,2fi​(⋅;θ)≡fi​(⋅−θ) , i=1,2 , θ∈R\theta\in\mathbb{R}θ∈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)α∈(0,1). Moran (1973) suggested a test which is based on a single split of the data, \textit{i.e.,} to use X2X_2X2​ in order to conduct a one-sided test in the direction of X1X_1X1​. Specifically, if b1b_1b1​ and b2b_2b2​ are the (1−α)(1-\alpha)(1−α)'th and α\alphaα'th quantiles associated with the distribution of X2X_2X2​ under HHH, 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 a∈Ra\in\mathbb{R}a∈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)f1​(⋅) and f2(⋅)f_2(\cdot)f2​(⋅) 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 ddd-dimensional Gaussians.

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