Abstract
It has long been customary to measure the adequacy of an estimator by the smallness of its mean squared error. The least squares estimators were studied by Gauss and by other authors later in the nineteenth century. A proof that the best unbiased estimator of a linear function of the means of a set of observed random variables is the least squares estimator was given by Markov [12], a modified version of whose proof is given by David and Neyman [4]. A slightly more general theorem is given by Aitken [1]. Fisher [5] indicated that for large samples the maximum likelihood estimator approximately minimizes the mean squared error when compared with other reasonable estimators. This paper will be concerned with optimum properties or failure of optimum properties of the natural estimator in certain special problems with the risk usually measured by the mean squared error or, in the case of several parameters, by a quadratic function of the estimators. We shall first mention some recent papers on this subject and then give some results, mostly unpublished, in greater detail.
This work was supported in part by an ONR contract at Stanford University.
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© 1992 Springer Science+Business Media New York
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James, W., Stein, C. (1992). Estimation with Quadratic Loss. In: Kotz, S., Johnson, N.L. (eds) Breakthroughs in Statistics. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0919-5_30
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DOI: https://doi.org/10.1007/978-1-4612-0919-5_30
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