Likelihood ratio tests and related confidence intervals for a real parameter in the presence of an infinite dimensional nuisance parameter are considered. In all cases, the estimator of the real parameter has an asymptotic normal distribution. However, the estimator of the nuisance parameter may not be asymptotically Gaussian or may converge to the true parameter value at a slower rate than the square root of the sample size. Nevertheless the likelihood ratio statistic is shown to possess an asymptotic chi-squared distribution. The examples considered are tests concerning survival probabilities based on doubly censored data, a test for presence of
heterogeneity in the gamma frailty model, a test for significance of the regression coefficient in Cox's regression model for current status data and a test for a ratio of hazards rates in an exponential mixture model. In both of the last examples the rate of convergence of the estimator of the nuisance parameter is less than the square root of the sample size.