Contributions on Theory of Mathematical Statistics by Kei Takeuchi

Author:Kei Takeuchi
Language: eng
Format: epub
ISBN: 9784431552390
Publisher: Springer Japan

(8.7)

where the coefficients s are dependent on J and

We call such an estimator to be a linear estimator for .

Arranging all possible Js, we can assign a number to each J and a linear estimator can be written as and

(8.8)

From linear independence of s unbiasedness implies that

(8.9)

Thus a linear unbiased estimator can be identified with a set of s satisfying the two conditions (8.8) and (8.9). Also the variance of it can be written as

Hence to make the variance small, we should choose the coefficients ’s so as to make as small as possible.

However since depends on the parameter , there is usually no uniformly best choice of s, so that we shall consider an estimator which makes the expected variance with respect to some prior distribution over the parameter space to be as small as possible. Let be a prior distribution over (generally there is no -field defined over , so general definition of is impossible; we think that is defined over finite set of points and ) and denote . Let and seek for s which make Q minimum satisfying (8.8) and (8.9). Omitting all s which are predetermined to be equal to zero, the problem reduces to minimizing with condition , where the symbol implies the sum over non- predetermined s. Introducing Lagrangian multipliers and differentiating the Lagrangian form we have

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