On the rate of convergence in the central limit theorem for signed rank statistics
This paper deals with the rate of convergence of the distribution functions of signed rank statistics for one sample to the normal d.f.: \[ S^+_ N=N^{-1}\sum^{N}_{i=1}1_{[0,\infty [}(X_ i)\cdot \Psi (R_ i/N+1) \] where \(R_ i\) is the rank of \(| X_ i|\) among \(| X_ 1|,...,| X_ N|.\) It generalizes known results to non-standard cases: when the underlying observations are independent but not identically distributed, and for score generating functions \(\Psi\) which are unbounded and not necessarily monotonous (including the normal scores statistic). The approach is the same as the Chernoff-Savage, expressing the statistic \(S^+_ N\) as an integral of the empirical distribution functions; so, they state a generalization of Dvoretzky-Kiefer-Wolfovitz' theorem to the non identically distributed case. As a corollary of the proof, a strong law for \(S^+_ N\) and a functional central limit theorem are given.
distribution functions of signed rank statistics
integral of the empirical distribution functions
generalization of Dvoretzky-Kiefer- Wolfovitz' theorem
scientific article; zbMATH DE number 3907555
scientific article