A New Way to Derive Locally Most Powerful Rank Tests

Abstract

The standard way to derive locally most powerful rank tests in nonparametrics involves differentiation of the rank likelihood, which is a complicated integral, and the proof is typically quite long and technical. By treating the ranks as the observed data and the underlying continuous random variables as the “complete” data, we are able to give a quick derivation of locally most powerful rank tests by using a well-known relationship between the derivatives of the log-likelihoods of the observed and “complete” data, which can be either assumed known or derived from first principles in a few lines. Along with shortening the proof considerably, this proposed approach has two other advantages. First, it allows students to see that the locally most powerful rank test is just a special case of the locally most powerful test based on the efficient score (i.e., the derivative of the log-likelihood), a topic often covered in a first-year graduate course in mathematical statistics. Second, it allows students to gain the perspective of viewing a rank test as an case of inference based on incomplete data. As a novel application, the proposed approach is used to derive a new rank test based on the difference between the Savage score and the Wilcoxon score. The new test is useful in testing for a difference in two survival distributions with a time-varying hazard ratio.

Keywords:  :

• Complete data
• Hazard ratio
• Locally most powerful test
• Savage test
• Score function
• Wilcoxon test