Sufficiency and Invariance

Summary

Let (X, A, P)be a given statistical model and letg be the class of all one-to-one, bimeasurable maps g of ( X, A) onto itself such that g is measure-preserving for each P ε P, i.e. Pg⁻¹= P for all P. Let us suppose that there exists a least (minimal) sufficient sub-fieldL. Then, for each LεL, it is true that g⁻¹L is P-equivalent to L for each gεg , i.e., the least sufficient sub-field is almost g -invariant. It is demonstrated that, in many familiar statistical models, the least sufficient sub-field and the sub-field of all almost g -invariant sets are indeed P-equivalent. The problem of data reduction in the presence of nuisance parameters has been discussed very briefly. It is shown that in many situations the principle of invariance is strong enough to lead us to the standard reductions. For instance, given n independent observations on a normal variable with unknown mean (the nuisance parameter) and unknown variance, it is shown how the principle of invariance alone can reduce the data to the sample variance.