On the relative error in normal approximation for signed rank statistics with a broad range of scores

Abstract

The Cramér-type large deviation results in the optimal range of 0<x⩽o(N14) are obtained on the relative error in normal approximation for the signed rank statistics under the symmetry hypothesis. The methods used are applications of elementary conditional probability and Feller's Theorem 1 [Trans. Amer. Math. Soc. 54 (1943), 361–372] (as Kallenberg, Z. Wahrsch. Verw. Geb. 60 (1982), 403–409, did to obtain the optimal range of 0<x⩽o(N16) for the simple linear rank statistics). The results obtained are valid with a broad range of regression constants and scores (allowed to be generated by discontinuous functions, but not necessarily) restricted by mild conditions, while most of previous results for the problem dealing with rank statistics are obtained with a narrow range of scores generated by differentiable and bounded functions, and/or with severely restricted regression constants.