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以可容许性为试金石AI 解读

Consider the problem of estimating simultaneously the means $\theta_i$ of independent normal random variables $x_i$ with unit variance. Under the weighted quadratic loss $L(\theta, a) = \sum_i\lambda_i(\theta_i - a_i)^2$ with positive weights it is well known that: (1) An estimator which is admissible under one set of weights is admissible under all weights. (2) Estimating individual coordinates by proper Bayes estimators results in an admissible estimator. (3) Estimating…

希尔伯特空间高斯分布均值的全部可容许线性估计量AI 解读

We consider linear estimators for the mean $\theta$ of a Gaussian distribution $N(\theta, C)$ on a Hilbert space, when the covariance operator $C$ is known. It was argued in a previous work that the natural class of linear estimators is the class of measurable linear transformations. Using the simplest quadratic loss we prove that the linear estimator $L$ is admissible if and only if the operator $C^{-1/2}LC^{1/2}$ is Hilbert-Schmidt, self-adjoint, its eigenvalues are all…

对称统计量的不变性原理AI 解读

We derive invariance principles for processes associated with symmetric statistics of arbitrary order. Using a Poisson sample size, such processes can be viewed as functionals of a Poisson Point Process. Properly normalized, these functionals converge in distribution to functionals of a Gaussian random measure associated with the distribution of the observations. We thus obtain a natural description of the limiting process in terms of multiple Wiener integrals. The results…