The Equivariance Criterion in Statistical Prediction and Its Ramifications
Open AccessConsider predicting an unobserved random variable Y based on the observed value of a random vector X, under an assumed model. In this context, we formalize the basic logic of equivariance to serve it as a useful tool for selecting a predictor. The development involves a new framework and more intricate transformation groups, compared to those involved in a standard decision problem. We establish the structure and many interesting properties of the transformation groups. Conditions for the existence of equivariant predictors, construction of all equivariant predictors and determination of a uniformly minimum risk equivariant predictor are described in terms of maximal invariants and orbits. The theoretical results are applied to location, scale and location-scale models and illustrated with several examples. We prove that a unique best risk-unbiased predictor, if it exists, is equivariant, and under certain conditions, a best equivariant predictor is risk-unbiased. We also show that a prediction problem can be reduced to a decision problem, involving only the random observable X and its marginal model, with an appropriate treatment of the decision space. In particular, invariance of the original problem conveys to the reduced problem, and for each equivariant predictor in the original problem, there exists an equivariant decision rule in the reduced problem with the same risk function. Finally, we combine equivariance with the Pitman closeness criterion and derive optimal estimators and predictors for location, scale and location-scale models. This approach is very robust with respect to the choice of the loss function, as the optimality of best rules holds for a wide class of loss functions. These results are also applied to several practical problems.
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