Three Scores and 15 Years (1948-2023) of Rao's Score Test: A Brief History

TL;DR

This paper reviews the history and development of Rao's score test from its inception in 1948 to its current status, highlighting its significance and recent research directions in statistics and econometrics.

Contribution

It provides a comprehensive historical overview of Rao's score test, emphasizing its evolution, interpretation of earlier tests, and recent advancements in robust hypothesis testing.

Findings

Rao's score test became a fundamental hypothesis testing tool.

Historical tests can be interpreted as special cases of the RS test.

Recent research focuses on robustness under model misspecification.

Abstract

Rao (1948) introduced the score test statistic as an alternative to the likelihood ratio and Wald test statistics. In spite of the optimality properties of the score statistic shown in Rao and Poti (1946), the Rao score (RS) test remained unnoticed for almost 20 years. Today, the RS test is part of the ``Holy Trinity'' of hypothesis testing and has found its place in the Statistics and Econometrics textbooks and related software. Reviewing the history of the RS test we note that remarkable test statistics proposed in the literature earlier or around the time of Rao (1948) mostly from intuition, such as Pearson (1900) goodness-fit-test, Moran (1948) I test for spatial dependence and Durbin and Watson (1950) test for serial correlation, can be given RS test statistic interpretation. At the same time, recent developments in the robust hypothesis testing under certain forms of misspecification, make the RS test an active area of research in Statistics and Econometrics. From our brief account of the history the RS test we conclude that its impact in science goes far beyond its calendar starting point with promising future research activities for many years to come.