Sequential signed rank statistics

Abstract

In order to adapt rank methods to sequential analysis, we consider one-sample rank statistics which are built upon sequential instead of ordinary ranks. As both types of rank statistics are asymptotically equivalent within local models, we have to study their relative performance under a fixed distribution. This is exemplified by means of repeated significance tests (thereby making use of an idea borrowed from renewal theory). A function¬al limit theorem and a SLLN are derived as the main tools for any such study. In order to get the first one, we employ a refinement of the projection method and a decomposition by which a rank statistic is presented as a sum of a martingale, an inverse martingale and a linear function of order statistics. This decom¬position also yields a SLLN. We prefer a direct argument, however, which can do with much weaker regularity conditions.

Keywords:

• sequential ranks
• functional limit theorems
• SLLN
• repeated significance tests
• CLT for stopping times
• sequential ARE under fixed alternatives
• Bahadur representation of LFOS