We show that the asymptotic mean of the log-likelihood ratio in a misspecified model is a differential geometric quantity that is related to the exponential curvature of Efron (1978), Amari (1982), and the preferred point geometry of Critchley et al. (1993, 1994). The mean is invariant with respect to reparametrization, which leads to the differential geometrical approach where coordinate-system invariant quantities like statistical curvatures play an important role. When models are misspecified, the likelihood ratios do not have the chi-squared asymptotic limit, and the asymptotic mean of the likelihood ratio depends on two geometric factors, the departure of models from exponential families (i.e. the exponential curvature) and the departure of embedding spaces from being totally flat in the sense of Critchley et al. (1994). As a special case, the mean becomes the mean of the usual chi-squared limit (i.e. the half of the degrees of freedom) when these two curvatures vanish. The effect of curvatures is shown in the non-nested hypothesis testing approach of Vuong (1989), and we correct the numerator of the test statistic with an estimated asymptotic mean of the log-likelihood ratio to improve the asymptotic approximation to the sampling distribution of the test statistic.
differential geometry log-likelihood ratio asymptotic mean exponential curvature preferred point geometry non-nested hypothesis