Notions of Limiting P Values Based on Data Depth and Bootstrap

In this article we propose some new notions of limiting P values for hypothesis testing. The limiting P value (LP) here not only provides the usual attractive interpretation of a P value as the strength in support of the null hypothesis coming from the observed data, but also has several advantages. First, it allows us to resample directly from the empirical distribution (in the bootstrap implementations), rather than from the estimated population distribution satisfying the null constraints. Second, it serves as a test statistic and as a P value simultaneously, and thus enables us to obtain test results directly without having to construct an explicit test statistic and then establish or approximate its sampling distribution. These are the two steps generally required in a standard testing procedure. Using bootstrap and the concept of data depth, we have provided LP's for a broad class of testing problems where the parameters of interest can be either finite or infinite dimensional. Some computer simulation results show the generality and the computational feasibility of our approach.