Abstract
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.
Highlights
The score test proposed by Rao (1947) has been widely used in the recent years for data analysis and model building
The test statistics have been denoted as Ws1 andWs2 and have the same forms as WS but the parameters p and λ in I are replaced by p0 andfor Ws1 and by pandfor Ws2 respectively. It has been noted by several authors (Cox and Snell (1989), Davison (2003), Morgan (2000), Pawitwan (2001), Morgan et al (2007), Verbeke and Molenberghs (2007) and Freedman (2007)) that observed Fisher information matrix may be used in place of the expected Fisher information matrix in a score test statistic
We observed that the power functions show fluctuations when the test statistics become negative and lead to the acceptance of the null hypothesis
Summary
The score test proposed by Rao (1947) has been widely used in the recent years for data analysis and model building. Freedman (2007) discussed some of the theoretical reasons for the inconsistency of the score test He observed that the test was inconsistent when the observed Fisher information matrix was used rather than the expected Fisher information matrix and that the observed Fisher information matrix generated negative variance estimates at the maximum likelihood estimate (MLE) of the parameter of the distribution under the null hypothesis. It has been found that the usual score test statistic is more consistent than its perturbed version obtained by using the unrestricted MLEs in the expected or the observed Fisher information matrix This finding contradicts the conclusion of Freedman (2007) where the asymptotic properties were of major concern.
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