A Note on the Wilcoxon Score for the One-Sample Problem

In this study, we consider to unify the sign and signed-rank tests for the one-sample problem by introducing the score functions. In this approach, we propose a generalization of the signed-rank statistics for the one-sample case. Then we derive the asymptotic normality of the proposed statistics using the large sample approximation theorem for the real application to the problem. For this purpose, we use the Liapounov central limit theorem, which requires the higher moments than the second moment. Then we compare the efficiency between the proposed test and the existed rank-sum tests by using the generalization form of the score function by obtaining empirical powers through a simulation study. We consider the three different sample sizes and four kinds of underlying distributions for this comparison study. Finally, we discuss the some useful aspect of median and comment briefly on the resampling methods for the derivation of the null distributions for any given nonparametric statistics.