Nonuniform Rates of Convergence to Normality for Two-Sample U-Statistics in Non IID Case with Applications

Abstract

Rates of convergence to normality are studied for two-sample U-statistics in non iid case under certain conditions which ensures that all moments of the kernel exist but the moment generating function of the kernel may not exist. Applications are made to compute normal approximation zone for the tail probability, nonuniform L _p version of Berry–Esseen theorem and moment type convergences of a standardized U-statistic. The normal approximation zone goes beyond moderate deviation and extends up to large deviation. As an application, efficiency of U-statistic-based tests, when the basic observations are discretized, is studied. It is seen that sometimes test efficiency may increase after discretization. Possible explanation is provided for such intriguing phenomena. A further application is made of deviation probabilities to compare agricultural production scenarios, e.g., growth of Elephant-foot-yam, over years. Yam stem growth is a good predictor for underground yam deposition. A nonparametric robust procedure to estimate derivative of a function based on discrete data is proposed. Performance of the proposed technique that is insensitive to outliers is investigated and found to be satisfactory. The procedure of analysis adopted in yam data may be extended to the cases where variables of interest are continuous growth curves that need to be compared over different time cycles in terms of rare events.