Tail probability approximation for one and two sample U-statistics by the saddlepoint method

This dissertation is devoted for proposing an alternative to normal (large sample) approximation for the tail probability approximation (under both null and alternative hypothesis) of a one and two sample U-Statistics by saddlepoint methods.;Under minimal assumptions, approximations for the Cumulant Generating Function of both one and two sample standardized U-Statistic are derived by using Hoeffding's decomposition. These approximations are then used in Daniel's (1987) formulation (for a discrete case) of Lugannani and Rice's (1980) saddlepoint approximation to the tail probability.;This method has been applied to both Wilcoxon Signed Rank test and the Mann-Whitney form of Wilcoxon's Rank Sum test to approximate the tail probability under null and various shift (normal, logistic and Cauchy) alternatives. Finally, results using our approximation have been compared to both normal approximation and exact distribution (based on 10 million simulations) for both the examples.