The paper "Estimation with Quadratic Loss" by W. James and C. Stein discusses the properties of estimators in problems where the risk is measured by the mean squared error or a quadratic function of the estimators. The authors review previous work on this topic, including studies by Gauss, Markov, David, Neyman, Aitken, Fisher, Pitman, Wald, Blackwell, Hodges, Lehmann, Blyth, Girshick, Savage, Karlin, and themselves. They present new results, particularly in the case of multivariate normal distributions, and provide detailed proofs for some of their findings. Key points include: - The inadmissibility of the usual estimator for the mean of a multivariate normal distribution with at least three dimensions. - The admissibility of certain estimators in low-dimensional cases, such as one or two location parameters. - Conditions under which Pitman's estimator is admissible or inadmissible for estimating multiple location parameters. - The minimax property of the natural estimator in certain problems, but its inadmissibility in others. - Examples where the natural estimator is not even minimax, despite having constant risk. The paper also addresses open problems and conjectures, particularly regarding the admissibility of estimators for location parameters and the behavior of the natural estimator in more complex problems.The paper "Estimation with Quadratic Loss" by W. James and C. Stein discusses the properties of estimators in problems where the risk is measured by the mean squared error or a quadratic function of the estimators. The authors review previous work on this topic, including studies by Gauss, Markov, David, Neyman, Aitken, Fisher, Pitman, Wald, Blackwell, Hodges, Lehmann, Blyth, Girshick, Savage, Karlin, and themselves. They present new results, particularly in the case of multivariate normal distributions, and provide detailed proofs for some of their findings. Key points include: - The inadmissibility of the usual estimator for the mean of a multivariate normal distribution with at least three dimensions. - The admissibility of certain estimators in low-dimensional cases, such as one or two location parameters. - Conditions under which Pitman's estimator is admissible or inadmissible for estimating multiple location parameters. - The minimax property of the natural estimator in certain problems, but its inadmissibility in others. - Examples where the natural estimator is not even minimax, despite having constant risk. The paper also addresses open problems and conjectures, particularly regarding the admissibility of estimators for location parameters and the behavior of the natural estimator in more complex problems.