Asymptotic distribution of rank statistics under dependencies with multivariate application (Q753322)

[X\({}_ n=(X_{1n},...,X_{Jn}),1\leq n\leq N]\) is a set of independent random vectors of length J, the marginal cdfs being absolutely continuous. Define \(u(x)=1\) or 0 according to whether \(x\geq 0\) or \(x<0\); define \(s(x)=1\), 0, or -1, according to whether \(x>\), \(=\), or \(<0\). Define \[ R_{jn}=\sum^{J}_{b=1}\sum^{N}_{c=1}u(X_{jn}- X_{bc}),\text{ and } R^+_{jn}=\sum^{J}_{b=1}\sum^{N}_{c=1}u(| X_{jn}| - | X_{bc}|). \] Let \(\phi\) be a real-valued nonzero function defined on (0,1), and define \(a_ M(m)\) as \(\phi (m/(M+1))\). Define \[ S_ N=\sum^{J}_{j=1}\sum^{N}_{n=1}d_{jn}a_ M(R_{jn})\text{ and } S^+_ N=\sum^{J}_{j=1}\sum^{N}_{n=1}d_{jn}s(X_{jn})a_ M(R^+_{jn}), \] where \([d_{jn}]\) are arbitrary constants not all equal to zero. Conditions are given under which \(S_ N\) and \(S^+_ N\) are asymptotically normal. The results are used to construct a test of the hypothesis that the marginal medians in a multivariate distribution are all equal to zero.