Local Bahadur optimality of some rank tests of independence
type: Item
instance of: scholarly article
title: Local Bahadur optimality of some rank tests of independence
zbMATH Open document ID: 0646.62038
DOI: 10.1016/0167-7152(87)90101-5
author: Piotr Bajorski
published in: Statistics \& Probability Letters
publication date: 1987-01-01T00:00:00Z
review text: Let \((X_ 1,Y_ 1),...,(X_ n,Y_ n)\) be independent r.v.'s with common continuous bivariate distribution function (d.f.) H and marginal d.f.'s F and G. To test the independence hypothesis, \[ {\mathcal H}_ 0=\{H_ 0: H_ 0(x,y)=F(x)G(y),\text{ F and G continuous d.f.'s on }{\mathbb{R}}\}, \] against positive quadrant dependence, \textit{T. Ledwina}, Sankhyā, Ser. A 48, 188-207 (1986; Zbl 0606.62046), has introduced tests of independence based on the statistics, \[ t_ n=\sup_{0\leq p\leq 1}n^{1/2}\bar t_ n(p),\quad t_ n^{\epsilon}=\sup_{\epsilon \leq p=1-\epsilon}(p(1-p))^{-1/2}n^{1/2}\bar t_ n(p),\quad \epsilon \in (0,1/2) \] where \(\bar t{}_ n(p)=n^{-2}\sum^{n}_{i=1}R_ i\) \((p-I(S_ i\leq np))\), I(E) is the indicator function of the set E, while \(R_ 1,...,R_ n\) and \(S_ 1,...,S_ n\) are the ranks of \(X_ 1,...,X_ n\) and \(Y_ 1,...,Y_ n\), respectively. The author studies the local optimality, in the Bahadur sense, of \(t_ n\) and \(t^{\epsilon}_ n\) for testing \({\mathcal H}_ 0\) against a class of positively quadrant dependent distributions. He finds the classes of distribution functions for which the tests based on \(t_ n\), \(t_ n^{\epsilon}\) and some linear rank tests are locally optimal. It is also proved that \(t_ n\) and \(t_ n^{\epsilon}\) are at least as efficient as the respective linear rank statistics.
Mathematics Subject Classification ID: 62G10
zbMATH DE Number: 4054816
zbMATH Keywords: sufficient and necessary conditions for local Bahadur optimality
zbMATH Keywords: local comparisons of tests
zbMATH Keywords: Bahadur slope
zbMATH Keywords: Bahadur efficiency
zbMATH Keywords: Kullback-Leibler information
zbMATH Keywords: positive quadrant dependence
zbMATH Keywords: tests of independence
zbMATH Keywords: linear rank statistics
reviewed by: Aldo J. Viollaz
MaRDI profile type: Publication
full work available at URL: 0167-7152(87)90101-5
OpenAlex ID: W2066459229
cites work: Rates of Convergence of Estimates and Test Statistics
cites work: The performance of some correlation coefficients for a general bivariate distribution
cites work: Local Bahadur efficiency of rank tests for the independence problem
cites work: Q3745065
cites work: On the limiting Pitman efficiency of some rank tests of independence
cites work: An expansion of the index of large deviations for linear rank statistics
cites work: Q5629057
cites work: Q3323004
cites work: Large Deviations and Bahadur Efficiency of Linear Rank Statistics
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description: scientific article; zbMATH DE number 4054816