J. Japan Statist. Soc., Vol. 31 (No. 1), pp. 111-127, 2001

The Lower Bound for MSE in Statistical Prediction Theory

Yoichi Miyata

Abstract. This paper is concerned with an inequality for MSE in statistical prediction theory. Takeuchi (1975) provided the inequality for a risk of unbiased predictor under certain regularity conditions. We shall provide an inequality for MSE of an unbiased predictor from L2-differentiability of densities point of view. In addition, this inequality is simplified and corresponded to the above under slightly stronger conditions. We shall also state the criterion for L2-differentiability in the case that an observable random vector and a predictive random variable are not independent.

Key words and phrases: unbiased predictor, Cramér-Rao inequality, differentiability in quadratic mean.

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