On the approximate elimination of nuisance parameters by conditioning

SUMMARY: The general problem of inference about a scalar parameter of interest θ in the presence of a nuisance parameter λ using conditional inference is considered.xml. A condition is given under which inference based on the conditional distribution of θ, the maximum likelihood estimate of θ, given λ_o, the maximum likelihood estimate of λ for fixed θ = θ₀, is optimal, in a certain sense. When this condition is not satisfied, it is shown that inference should be based on the conditional distribution of θ given λ_o, j_o where j_o denotes the observed information for λ for fixed θ =θθ_o, although this will involve some loss of information about θ. This information loss is shown to be related to the statistical curvature of the model.