Estimation of the Mean of a Multivariate Normal Distribution

By C. Stein. 1981

Subfields: Statistics and Probability, Probabilistic & Bayesian Methods

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Citations in the corpus

Kernel Mean Estimation via Spectral Filtering (2014, methodology): “The proposed estimators are simple to implement and perform well in practice.”
Verified Uncertainty Calibration (2019, background): “Bias is a common issue with statistical estimators, for example, the seminal work by Stein [43] fixes the bias of the mean-squared error.”
Sliced Score Matching: A Scalable Approach to Density and Score Estimation (2019, methodology): “Two prior methods for score estimation are based on a generalized form of Stein’s identity (Stein, 1981; Gorham & Mackey, 2017): where q ( x ) is a continuously differentiable density and h ( x ) is a function satisfying some regularity conditions.”
Nearly unbiased variable selection under minimax concave penalty (2010, methodology): “The continuity of β̂, demanded by Stein (1981), is a property of independent interest on its own right for robust estimation [Fan and Li (2001)].”
Provably Robust Deep Learning via Adversarially Trained Smoothed Classifiers (2019, background): “The equality (a) is known as Stein’s lemma [31], although we note that something similar can be derived for more general distributions.”
Stein Unbiased GrAdient estimator of the Risk (SUGAR) for Multiple Parameter Selection (2014, methodology): “For a fixed θ, the celebrated Stein’s lemma [62] allows to unbiasedly estimate R{μ}(μ0, θ) through the weak Jacobian ∂1μ(y, θ), where the subscript 1 specifies that the (weak) Jacobian is with respect to the first argument y.”
A Spectral Approach to Gradient Estimation for Implicit Distributions (2018, background): “Central to these works is an equation that generalizes the original Stein’s identity (Stein, 1981), shown in the following theorem.”
How to Train Your Energy-Based Models (2021, background): “Similar formulations are also explored by Raphan and Simoncelli (2007, 2011) and can be traced back to Tweedie’s formula (1956) and Stein’s Unbiased Risk Estimation (Stein, 1981).”
Denoising of 3 D Magnetic Resonance Images by Using 1 Higher-Order Singular Value Decomposition 2 (2014, background): “16 Stein's unbiased risk estimate (Stein, 1981) or the no-reference metric Q (Zhu and 17 Milanfar, 2010) can serve as quantitative metrics for automatic parameter tuning 18 when no "ground-truth" reference is available, but will require a number of iterations.”
Denoising of 3D magnetic resonance images by using higher-order singular value decomposition (2015, background): “Stein’s unbiased risk estimate (Stein, 1981) or the no-reference metric Q (Zhu and Milanfar, 2010) can serve as quantitative metrics for automatic parameter tuning when no ‘‘ground-truth’’ reference is available, but will require a number of iterations.”
Sharp adaptation for inverse problems with random noise (2002, background): “A well-known idea of choosing ~ h k is based on the unbiased estimation of the risk (Mallows (1973), Akaike (1973) and Stein (1981)).”
Handbook of statistical distributions with applications (2006, unknown): “Stein’s (1981) Lemma: If X follows a normal distribution with mean μ and standard deviation σ, then E(X − μ)h(X) = σE [ ∂h(X) ∂X ]”
Risk hull method and regularization by projections of ill-posed inverse problems (2006, unknown): “The Annals of Statistics 2006, Vol. 34, No. 4, 1653–1677 DOI: 10.1214/009053606000000542 © Institute of Mathematical Statistics, 2006”
Neural Empirical Bayes (2019, background): “ random variable) and X(for the noisy/measured random variable) used in (Robbins, 1955). There is also a deep connection between Miyasawa’s least squares estimator and Stein’s unbiased risk estimate (Stein, 1981), where again for Gaussian noise the risk estimate has a closed form as expectations ove”
A NOTE ON THE LASSO AND RELATED PROCEDURES IN MODEL SELECTION (2006, methodology): “In practice, since β is unknown, several methods, such as generalized cross validation (Craven and Wahba (1979)), k-fold cross validation, or Stein’s unbiased estimate of risk (Stein (1981)), can be used for the purpose of minimizing the squared error.”