A computable confidence upper limit from discrete data with good coverage properties

Abstract

We present a new and simple method for constructing a 1− α upper confidence limit for θ in the presence of a nuisance parameter vector ψ, when the data is discrete. Our method is based on computing a P-value P{ T⩽ t} from an estimator T of θ, replacing the nuisance parameter by the profile maximum likelihood estimate ψ ̂ (θ) for θ known, and equating to α. We provide a theoretical result which suggests that, from the point of view of coverage accuracy, this is close to the optimal replacement for the nuisance parameter. We also consider in detail limits for the (i) slope parameter of a simple linear logistic regression, (ii) odds ratio in two-way tables, (iii) ratio of means for two Poisson variables. In all these examples the coverage performance of our upper limit is a dramatic improvement on the coverage performance of the standard approximate upper limits considered.