The generalised problem of the Nile: robust confidence sets for parametric functions

The pivotal model is described and applied to the estimation of parametric functions ϕ(θ). This leads to equations of the form H(x;θ)=G{p(x,θ)}. These can be solved directly or by the use of differential equations. Examples include various parametric functions ϕ(θ,σ) in a general location-scale distribution f(p),p=(x−θ)/σ and in two location-scale distributions. The latter case includes the ratio of the two scale parameters σ1/σ2, the difference and ratio of the two location parameters θ1−θ2 and the common location θ when θ1=θ2=θ. The use of the resulting pivotals to make inferences is discussed along with their relation to examples of non-uniqueness occurring in the literature.