In this chapter we consider the traditional type of nuisance parameter model in which, in addition to the parameter θ we also have a vector of nuisance parameters (ξ1,…,ξp) such that θ, ξ1,…,ξPis a complete description of the probability model. By this we mean that if θ(P)=θ(Q) and ξi(P)=ξi(Q) for i=1,…,p then P=Q. The primary problem that we shall discuss is the construction of a space Ψ with constant covariance structure for different nuisance parameter problems. In section 4.1, we consider the use of group invariant methods to eliminate nuisance parameters with particular attention to location-scale models. In section 4.2 the use of conditional models is described in the context of inference functions. Finally, in section 4.3, we shall consider the construction of inference functions in more difficult situations where techniques described earlier do not work.
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