Invariance of a statistical procedure

The equivariance (see below) of some decision rule in a statistical problem, the statement of which admits of a group of symmetries, under this group . The notion of invariance of a statistical procedure arises in the first instance in so-called parametric problems of mathematical statistics, when there is a priori information: the probability distribution of the outcomes of an observation belongs to a known family . A statistical decision problem is said to be -equivariant under a group of measurable transformations of a measurable space of outcomes if the following conditions hold: 1) there is a homomorphism of onto a group of transformations of the parameter space ,

with the property

2) there exists a homomorphism of onto a group of measurable transformations of a measurable space of decisions ,

with the property

where is the loss function; and 3) all the additional a priori information on the possible values of the parameter (the a priori density , the subdivision into alternatives , etc.) is -invariant or -equivariant. Under these conditions, the decision rule , whether deterministic or random, is called an invariant (more precisely, a -equivariant) procedure if

The risk

of an equivariant decision procedure is -invariant; in particular, it does not depend on if the group acts transitively on .

In parametric problems there is, in general, no guaranteed optimal decision procedure which minimizes the risk for each value of the parameter . In particular, a procedure may lead to very small values of the risk for certain values of at the expense of worsening the quality for other equally-possible a priori values of the parameter. Equivariance guarantees to some extent that the approach is unbiased. When the group is sufficiently rich, there is an optimal invariant procedure with a uniformly minimal risk among the invariant procedures.

Invariant procedures are widely applied in hypotheses testing (see also Invariant test) and in the estimation of the parameters of a probability distribution. Thus, in the problem of estimating an unknown vector of means for the family of -dimensional normal distributions

with unit covariance matrix and quadratic loss function , the optimal equivariant estimator is the ordinary sample mean

Here the group is given by the product of the group of permutations of the observations and the group of motions of the Euclidean space ; . For , there exist for this problem non-equivariant estimators leading to a smaller risk than for for all ; however, the region of essential "superefficiency" turns out to be insignificant and diminishes without bound as the size of the sample increases. The possibility of superefficient procedures is connected with the non-compactness of .

Equivariant statistical procedures also arise in a number of non-parametric statistical problems, when the a priori family of distributions of outcomes is essentially infinite-dimensional, as well as in the construction of confidence sets for the parameter of the distribution in the presence of nuisance parameters.

References

[1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)