Robust properties of likelihood ratio tests

The usual asymptotic chi-squared distribution for the likelihood ratio test statistic is based on the assumptions that the data come from the parametric model under consideration and that the parameter satisfies the null hypothesis. In this paper we examine the distribution of the likelihood ratio statistic when the data do not come from the parametric model, but when the ‘nearest’ member of the parametric family still satisfies the null hypothesis. In general, the likelihood ratio statistic no longer follows an asymptotic chi-squared distribution, and an alternative statistic based on the union-intersection approach is proposed.