A Note on the Mann-Whitney Statistic for Lehmann Alternatives

✍ Scribed by U. Abely

• Publisher: John Wiley and Sons
• Year: 1982
• Tongue: English
• Weight: 232 KB
• Volume: 24
• Category: Article
• ISSN: 0323-3847
• DOI: 10.1002/bimj.4710240605

✦ Synopsis

Abstract

For the case of the LEHMANN alternatives the paper presents some new facts on the MANN‐WHITNEY statistic and, in particular, its variance V(p, m, n), where p = P(x~i~<y~i~). Explicit formulas for U and V are used to prove, among other things, the following propositions: For any m, n, V is a one‐hump function of p, and the hump always lies in the interval (1/2(3 ‐ √5), 1/2(√5 ‐ 1)). If no restrictions are imposed on p the boundaries of this interval are sharp. Given s = m + n, V(1/2, s/2,s/2) is maximal among all values V(p, m, n). The formulas allow, moreover, the improvement of the known bounds for the variance of p̌ = U/mn.