On nonparametric profile analysis of several multivariate samples (Q1058240)

Let \((X_{it}^{(1)},...,X_{it}^{(p)})\), \(t=1,2,...,n_ i\) be a random sample from the ith population with p-variate continuous distribution function \(F_ i\) for \(i=1,2,...,k\). Let \(\{X_{it}^{(\alpha)}\), \(t=1,2,...,n_ i\), \(i=1,2,...,k\}\) be arranged in increasing order and denote by \(R_{it}^{(\alpha)}\) the rank of \(X_{it}^{(\alpha)}\) in this increasing sequence. Put \(F=(F_ 1,...,F_ k)\) and define \(\xi_{iu}^{(\alpha)}(F)\) by the probability that the \(u^{th}\) smallest in the ranking of \(\{X_{it}^{(\alpha)}\), \(t=1,2,...,n_ i\), \(i=1,2,...,k\}\) is from the \(i^{th}\) sample. The population profiles are said to be parallel with respect to the rank order of \(N=\sum^{k}_{i=1}n_ i\) observations if \(\xi_{iu}^{(1)}(F)=\xi_{iu}^{(2)}(F)=...=\xi_{iu}^{(p)}(F)\) for all \(u=1,2,...,N\) and \(i=1,2,...,k\). For a given system of scores \(\{s_ u\), \(u=1,2,...,N\}\), put \(\eta_ i^{(\alpha)}(F)=\sum^{N}_{u=1}s_ u\xi_{iu}^{(\alpha)}(F)/n_ i\). Then the \(i^{th}\) population profile is said to be parallel to the axis if \(\eta_ i^{(1)}(F)=\eta_ i^{(2)}(F)=...=\eta_ i^{(p)}(F).\) Asymptotically distribution-free tests for parallelism of profile for a subset of p variables and a subset of k populations are discussed, including linear rank statistics and generalized U-statistics. Their asymptotic powers and consistency of a class of tests are developed.