Confidence regions when the Fisher information is zero

We examine the asymptotic behaviour of confidence regions in identifiable one-dimensional parametric models with smooth likelihood function and information equal to zero at a critical point of the parameter space.Confidence regions are based on inversion of the likelihood ratio test statistic and of some common forms of the score and Wald test statistics. For fixed parameter values other than the critical point, all these statistics have limiting x^2(1) distributions, but for most of them the convergence is not uniform near the critical point. When it is not, confidence regions based on inverting the tests, using the x^2(1) approximation, do not asymptotically have the nominal level. The exception to this lack of locally uniform convergence occurs with the score test standardised by expected, rather than observed, information. For the regions based on the score test standardised by observed information and on the likelihood ratio test, conservative procedures that do not rely on the x^2(1) approximation can be developed, but they are much too conservative near the critical parameter value. The regions based on the Wald tests have asymptotic level less than 1/2, regardless of the procedure used. Our results suggest that no procedure based solely on the likelihood function will be satisfactory. Whether or not this is the case is an open problem. A simulation study illustrates the results of this paper.