SOME CONTRIBUTIONS TO THE THEORY OF UNBIASED STATISTICAL PREDICTION
Open AccessTwo research topics in statistical prediction, which can be regarded as a general inference problem, are discussed. One topic is finding and utilizing lower bounds for prediction mean squared error (MSE). We develop Kshirsagar type lower bounds for prediction MSE and discuss their usefulness. These bounds are shown to be at least as sharp as the corresponding Bhattacharyya bounds. As the regularity conditions of Bhattacharyya bounds are not required for the validity of the new lower bounds, our lower bounds are more widely applicable than Bhattacharyya bounds. By examining some sufficient conditions for attaining the lower bounds, we obtain a useful and easy method for finding minimum MSE unbiased predictors. The second topic is risk unbiasedness. We extend the concept of risk unbiasedness in statistical estimation to prediction and non-standard inference problems, by formalizing the idea that a risk unbiased predictor should be at least as close to the "true" predictant as to any "wrong" predictant, on the average. A novel aspect of our approach is measuring closeness between a predicted value and the predictant by a regret function, derived suitably from the given loss function. For squared error loss, we present a method for deriving best (minimum risk) risk unbiased predictors when the regression function is linear in a function of the parameters. We derive a Rao-Blackwell type result for a class of loss functions that includes squared error and LINEX losses as special cases. For location-scale families, we prove that if a unique best risk unbiased predictor exists, then it is equivariant. One important finding is that in some problems a best unbiased predictor does not exist, but a best risk unbiased predictor can be obtained. Thus, risk unbiasedness can be a useful tool for selecting a predictor.
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