Journal of Statistical Planning and Inference 10 (1984) 289-309 North-Holland 289 EDGEWORTH EXPANSIONS FOR SIGNED LINEAR RANK STATISTICS UNDER NEAR LOCATION ALTERNATIVES Madan L. PURI* Department of Mathematics, Indiana University, Bloomington IN 47405, USA Munsup SEOH Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, USA Received 3 February 1984 Recommended by 1. Vincze Abstract: Edgeworth expansions with the uniform remainder of order o(N-‘) are established for signed linear rank statistics with regression constants under near location alternatives. The results are obtained both with exact scores and with approximate scores, normalized with natural param- eters as well as with simple constants. AMS Subject Classification: Primary 62G10, 62020; Secondary 60F05. Key words and phrases: Edgeworth expansions; Signed linear rank statistics; Location alternatives. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1. Introduction Let XNl,XN2, . . . . X,, be independent rv’s (random variables) such that XNj= eNj+ q, 15 j< N, where the 0Nj’s are location parameters; { q},?=, is a sequence of iid rv’s (independently and identically distributed random variables) with a com- mon cdf (cumulative distribution function) F(x) and a pdf (probability density fun- tion)f(x). Let R&,, 1 sjrN, be the rank of [XNjI among {IX,,l: 1 rklN}. We consider the signed linear rank statistic T,$ = f cNja,,JR& sgn XNj j=l (1.1) where CNI, CN2? . . . , CNN are arbitrary KgreSSiOII COI’IScmS; aNI, aN2, . . . , aNN are scores; and sgn x= 1 or - 1 according as xr0 or xc0. We assume that the scores aNj, 1 IjlN, are generated by some known function J(t) (called a score generating function) defined on the open interval (0,l) in either *Research supported by the National Science Foundation grant. 0378-3758/84/$3.00 0 1984, Elsevier Science Publishers B.V. (North-Holland)