The paper discusses the estimation of parameters with quadratic loss, focusing on the admissibility of estimators. It begins by noting that the mean squared error is a common measure of estimator quality. The least squares estimator is well-known for its optimality in certain cases, but the paper explores situations where it is not optimal. In the case of multivariate normal distributions with identity covariance matrices, the usual estimator of the mean is shown to be inadmissible when the dimension is at least three. A modified estimator is introduced that has a lower risk. The paper also considers the case where the covariance matrix is unknown and provides results for estimators in such scenarios. The paper then formulates the general problem of admissible estimation with quadratic loss. It defines the risk function and discusses conditions under which an estimator is admissible. A theorem is presented that provides a sufficient condition for almost admissibility. The paper also examines the admissibility of Pitman's estimator for location parameters in low-dimensional cases. It shows that Pitman's estimator is admissible when the number of parameters is one or two but inadmissible when the number is three or more. The paper also discusses the case where nuisance parameters are present. In the case of multivariate analysis, the paper presents a problem where the natural estimator is not minimax. It shows that the natural estimator, which is invariant under the full linear group, is not minimax. The paper also discusses some unsolved problems in the field of estimation with quadratic loss, including the admissibility of estimators for location parameters and the behavior of estimators under more general loss functions.The paper discusses the estimation of parameters with quadratic loss, focusing on the admissibility of estimators. It begins by noting that the mean squared error is a common measure of estimator quality. The least squares estimator is well-known for its optimality in certain cases, but the paper explores situations where it is not optimal. In the case of multivariate normal distributions with identity covariance matrices, the usual estimator of the mean is shown to be inadmissible when the dimension is at least three. A modified estimator is introduced that has a lower risk. The paper also considers the case where the covariance matrix is unknown and provides results for estimators in such scenarios. The paper then formulates the general problem of admissible estimation with quadratic loss. It defines the risk function and discusses conditions under which an estimator is admissible. A theorem is presented that provides a sufficient condition for almost admissibility. The paper also examines the admissibility of Pitman's estimator for location parameters in low-dimensional cases. It shows that Pitman's estimator is admissible when the number of parameters is one or two but inadmissible when the number is three or more. The paper also discusses the case where nuisance parameters are present. In the case of multivariate analysis, the paper presents a problem where the natural estimator is not minimax. It shows that the natural estimator, which is invariant under the full linear group, is not minimax. The paper also discusses some unsolved problems in the field of estimation with quadratic loss, including the admissibility of estimators for location parameters and the behavior of estimators under more general loss functions.