On permutational central limit theorems for general multivariate linear rank statistics
Pranab Kumar Sen · NCSU Libraries Repository (North Carolina State University Libraries) · 1981
In nonparametric multivariate analysis, permutational central limit theorems (PCLT) play a vital role. In the context of multivariate multi sample rank order tests, Puri and Sen (1966, Theorem 4.1) considered a PCLT and this was later (Puri and Sen, 1969, Theorem 3.2) extended to general linear models. In either case, the proof provided by these authors is not correct (as will be explained in Section 2); nevertheless, the theorems remain valid under the stringent regularity conditions stated there. These theorems are multivariate generalizations of the classical (univariate) PCLT, which in its most general form is due to H?jek (1961). He was able to in corporate a powerful (quadratic mean) equivalence result for linear rank statistics and linear combinations of independent random variables (r.v.) which provide the desired result through the classical CLT. Puri and Sen (1971, Theorem 5.4.1) adapted this technique in the multivariate case; but, their result on the PCLT does not properly follow from the H?jek (1961) result, though the conclusion on the unconditional null distribution remains true. Conclusion on the PCLT in the multivariate case based on the moment convergence property (cf. Wald and Wolfowitz, 1944) is, of course, valid, but demands comparatively stringent regularity conditions. All these call for a re-examination of multivariate PCLT.