On the boundedness and nonmonotonicity of generalized score statistics

We show in the context of the linear regression model fitted by Gaussian quasi-likelihood estimation that the generalized score statistics of Boos and Hu and Kalbfleisch for individual parameters can be bounded and nonmonotone in the parameter, making it difficult tomake inferences from the generalized score statistic. The phenomenon is due to the form of the functional dependence of the estimators on the parameter being held fixed and the way this affects the score function and/or the estimator of the asymptotic variance. We note that in some settings, the score statistic can be bounded and nonmonotone.