Assume X1,...,Xn are i.i.d. according to a location scale model with distribution of the form F[(x − θ)/σ], where F is known, θ is a location parameter, and σ is a scale parameter. Suppose ˆθn is a location and scale equivariant estimator and ˆσn is a location invariant, scale equivariant estimator.
Then, show that the roots [ˆθn − θ]/σˆn and ˆσn/σ are pivots.