Local Bahadur optimality of some rank tests of independence
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.
sufficient and necessary conditions for local Bahadur optimality
scientific article; zbMATH DE number 4054816
scientific article