Improved minimax estimation of a multivariate normal mean under heteroscedasticity
Improved minimax estimation of a multivariate normal mean under heteroscedasticity
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异方差下多元正态均值的改进极小极大估计
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发表时间：
2015
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通讯作者：
Z. Tan
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作者：
Z. Tan
考虑用已知的方差阵估计多元正态均值的问题，它不一定与单位矩阵成正比。坐标与它们在Efron和Morris中的方差成正比缩小(J.Amer。统计学家。阿索克。68(1973)117-130)经验贝叶斯方法，而在Berger的(AN.统计学家。4(1976)223-226)极小极大估计。我们提出了一种新的极大极小估计量，通过在一类极大极小估计量中用正态先验近似最小化贝叶斯风险，其中收缩方向是开放的，并且收缩幅度被确定为达到极小极大。所提出的估计量具有一种有趣的简单形式，即一组坐标沿Berger估计量的方向收缩，其余的坐标沿贝叶斯规则的方向收缩。此外，所提出的估计器是尺度自适应的：它可以在规模类的正态先验(包括指定的先验)上同时达到接近最小贝叶斯风险，并且在相应的尺度类超矩形上达到接近最小的线性风险。在我们的数值研究中，对于不同的情形，所提出的具有极端先验的估计量比现有的极小极大估计量具有更大的风险降低。
Consider the problem of estimating a multivariate normal mean with a known variance matrix, which is not necessarily proportional to the identity matrix. The coordinates are shrunk directly in proportion to their variances in Efron and Morris’ (J. Amer. Statist. Assoc. 68 (1973) 117– 130) empirical Bayes approach, whereas inversely in proportion to their variances in Berger’s (Ann. Statist. 4 (1976) 223–226) minimax estimators. We propose a new minimax estimator, by approximately minimizing the Bayes risk with a normal prior among a class of minimax estimators where the shrinkage direction is open to specification and the shrinkage magnitude is determined to achieve minimaxity. The proposed estimator has an interesting simple form such that one group of coordinates are shrunk in the direction of Berger’s estimator and the remaining coordinates are shrunk in the direction of the Bayes rule. Moreover, the proposed estimator is scale adaptive: it can achieve close to the minimum Bayes risk simultaneously over a scale class of normal priors (including the specified prior) and achieve close to the minimax linear risk over a corresponding scale class of hyper-rectangles. For various scenarios in our numerical study, the proposed estimators with extreme priors yield more substantial risk reduction than existing minimax estimators.