A Note on inconsistency of the score test

The score test proposed by Rao (1947) has been widely used in the recent years for data analysis and model building because of its simplicity. However, at the time of its computation, it has been found that the value of the score test statistic becomes negative. Freedman (2007) discussed some of the theoretical reasons for this inconsistency of the score test and observed that the test was inconsistent when the observed Fisher information matrix was used rather than the expected Fisher information matrix. The present paper is an attempt to demonstrate the inconsistency of the score test in terms of the power function. The paper further dispels the doubt regarding the use of score test.