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Rao ’s ( 1948 ) seminal paper introduced a fundamental principle of testing based on the score function and the score test has local optimal properties. When the assumed model is misspecified, it is well known that Rao’s score (RS) test loses its optimality. A model could be misspecified in a variety of ways. In this paper, we consider two kinds: distributional and parametric. In the former case, the assumed probability density function differs from the data generating process.  Kent (1982)  and  White (1982)  analyzed this case and suggested a modified version of the RS test that involves adjustment of the variance. In the latter case, the dimension of the parameter space of the assumed model does not match with that of the true one. Using the distribution of the RS test under this situation,  Bera and Yoon (1993)  developed a modified RS test that is valid under the local parametric misspecification. This involves adjusting both the mean and variance of the standard RS test. This paper considers the  joint  presence of the distributional and parametric misspecifications and develops a modified RS test that is valid under both types of misspecification. Earlier modified tests under either type of misspecification can be obtained as the special cases of the proposed test. We provide three examples to illustrate the usefulness of the suggested test procedure. In a Monte Carlo study, we demonstrate that the modified test statistics have good finite sample properties.

Adjustments of Rao’s Score Test for Distributional and Local Parametric Misspecifications