Nonparametric tests for umbrella alternatives  SAM WU University of Florida Abstract: Some new distribution-free test procedures for umbrella alternatives will be presented. The tests extend the work of Jonckheere (1954) and Terpstra (1952) for ordered alternatives, Puri (1965), Mack and Wolf (1981) for umbrella alternatives (both known and unknown peak cases). Instead of averaging the pairwise Mann-Whitney statistics, we propose to combine their p-values using Fisher's combining functions. Monte Carlo power study showed that the new procedures improve substantially, especially for the unknown peak case. Asymptotic Pitman efficiency against translation alternatives has also been studied.