The Performance of the Likelihood Ratio Test When the Model is Incorrect

Let the random variable $X$ have a distribution depending on a parameter $\theta \in \Theta$. Consider the problem of testing the hypothesis $H: \Theta_0 \subseteqq \Theta$ based on a sequence of observations on $X$. The likelihood ratio test for $H$ is constructed by selecting a model for the unknown distribution of $X$. In this paper the asymptotic performance of the likelihood ratio test is studied when the model is incorrect, that is, when the probability distribution of $X$ is not a member of the model from which the likelihood ratio test is constructed. Exact and approximate measures of the asymptotic efficiency of the likelihood ratio test when the model is incorrect are proposed.