The “Three Plus One” Likelihood-Based Test Statistics: Unified Geometrical and Graphical Interpretations

Abstract

The presentations of the well-known likelihood ratio, Wald and score test statistics in textbooks appear to lack a unified graphical and geometrical interpretation. We present two simple graphical representations on a common scale for these three test statistics, and also the recently proposed gradient test statistic. These unified graphical displays may favor better understanding of the geometrical meaning of the likelihood-based statistics and provide useful insights into their connections.

• Statistical inference
• Geometrical interpretation
• Gradient statistic
• Graphical display
• Likelihood ratio
• Score
• Wald