丸山 祐造

The original Hotelling-Solomons inequality indicates that an upper bound of |mean - median|/(standard deviation) is 1. In this note, we find a new bound depending on the sample size, which is exactly smaller than 1.

Abstract We consider the estimation of the p-variate normal mean of $$X\sim \mathcal {N}_p(\theta ,I)$$ under the quadratic loss function. We investigate the decision theoretic properties of debiased shrinkage estimator, the estimator which shrinks towards the origin for smaller $$\Vert x\Vert ^2$$ and which is exactly equal to the unbiased estimator X for larger $$\Vert x\Vert ^2$$. Such debiased shrinkage estimator seems superior to the unbiased estimator X, which implies minimaxity. However, we show that it is not minimax under mild conditions.

This paper reviews minimax best equivariant estimation in these invariant<br /> estimation problems: a location parameter, a scale parameter and a (Wishart)<br /> covariance matrix. We briefly review development of the best equivariant<br /> estimator as a generalized Bayes estimator relative to right invariant Haar<br /> measure in each case. Then we prove minimaxity of the best equivariant<br /> procedure by giving a least favorable prior sequence based on non-truncated<br /> Gaussian distributions. The results in this paper are all known, but we bring a<br /> fresh and somewhat unified approach by using, in contrast to most proofs in the<br /> literature, a smooth sequence of non truncated priors. This approach leads to<br /> some simplifications in the minimaxity proofs.

This paper investigates estimation of the mean vector under invariant quadratic loss for a spherically symmetric location family with a residual vector with density of the form f (x, u) = eta((p+ n)/2) f (eta{parallel to x - theta parallel to(2) + parallel to u parallel to(2)}), where. is unknown. We show that the natural estimator x is admissible for p = 1, 2. Also, for p >= 3, we find classes of generalized Bayes estimators that are admissible within the class of equivariant estimators of the form {1 - xi(x/parallel to u parallel to)}x. In the Gaussian case, a variant of the James-Stein estimator, [1-{(p - 2)/(n+ 2)}/{parallel to x parallel to(2)/parallel to u parallel to(2) +(p - 2)/(n+ 2) + 1}]x, which dominates the natural estimator x, is also admissible within this class. We also study the related regression model.

We investigate Bayesian shrinkage methods for constructing predictive distributions. We consider the multivariate normal model with a known covariance matrix and show that the Bayesian predictive density with respect to Stein's harmonic prior dominates the best invariant Bayesian predictive density when the dimension is greater than or equal to 3. Alpha divergence from the true distribution to a predictive distribution is adopted as a loss function.

We consider quasi-admissibility/inadmissibility of Stein-type shrinkage estimators of the mean of a multivariate normal distribution with covariance matrix an unknown multiple of the identity. Quasi-admissibility/inadmissibility is defined in terms of nonexistence/existence of a solution to a differential inequality based on Stein's unbiased risk estimate (SURE). We find a sharp boundary between quasi-admissible and quasi inadmissible estimators related to the optimal James Stein estimator. We also find a class of priors related to the Strawderman class in the known variance case where the boundary between quasi-admissibility and quasi-inadmissibility corresponds to the boundary between admissibility and inadmissibility in the known variance case. Additionally, we also briefly consider generalization to the case of general spherically symmetric distributions with a residual vector. (C) 2017 Elsevier Inc. All rights reserved.

Zellner's g-prior is a popular prior choice for the model selection problems in the context of normal regression models. Wang and Sun [J. Statist. Plann. Inference 147 (2014) 95-105] recently adopt this prior and put a special hyper-prior for g, which results in a closed-form expression of Bayes factor for nested linear model comparisons. They have shown that under very general conditions, the Bayes factor is consistent when two competing models are of order 0 (n(tau)) for tau &lt; 1 and for tau =1 is almost consistent except a small inconsistency region around the null hypothesis. In this paper, we study Bayes factor consistency for nonnested linear models with a growing number of parameters. Some of the proposed results generalize the ones of the Bayes factor for the case of nested linear models. Specifically, we compare the asymptotic behaviors between the proposed Bayes factor and the intrinsic Bayes factor in the literature.

This paper studies Bayesian variable selection in linear models with general spherically symmetric error distributions. We construct the posterior odds based on a separable prior, which arises as a class of mixtures of Gaussian densities. The posterior odds for comparing among nonnull models are shown to be independent of the error distribution, if this is spherically symmetric. Because of this invariance, we refer to our method as a robust Bayesian variable selection method. We demonstrate that our posterior odds have model selection consistency, and that our class of prior functions are the only ones within a large class which are robust in our sense.

This work treats the problem of estimating the predictive density of a random vector when both the mean vector and the variance are unknown. We prove that the density of reference in this context is inadmissible under the Kullback-Leibler loss in a nonasymptotic framework. Our result holds even when the dimension of the vector is strictly lower than three, which is surprising compared to the known variance setting. Finally, we discuss the relationship between the prediction and the estimation problems.

Averaged orthogonal rotations of Zellner's g-prior yield general, interpretable, closed form Bayes factors for the normal linear model variable selection problem. Coupled with a model space prior that balances the weight between the identifiable and the unidentifiable models, limiting forms for the posterior odds ratios are seen to yield new expressions for high dimensional model choice.

We consider the problem of estimating the error variance in a general linear model when the error distribution is assumed to be spherically symmetric, but not necessary Gaussian. In particular we study the case of a scale mixture of Gaussians including the particularly important case of the multivariate-t distribution. Under Stein's loss, we construct a class of estimators that improve on the usual best unbiased (and best equivariant) estimator. Our class has the interesting double robustness property of being simultaneously generalized Bayes (for the same generalized prior) and minimax over the entire class of scale mixture of Gaussian distributions. (C) 2013 Elsevier B.V. All rights reserved.

This paper considers estimation of the predictive density for a normal linear<br /> model with unknown variance under alpha-divergence loss for -1 &lt;= alpha &lt;= 1.<br /> We first give a general canonical form for the problem, and then give general<br /> expressions for the generalized Bayes solution under the above loss for each<br /> alpha. For a particular class of hierarchical generalized priors studied in<br /> Maruyama and Strawderman (2005, 2006) for the problems of estimating the mean<br /> vector and the variance respectively, we give the generalized Bayes predictive<br /> density. Additionally, we show that, for a subclass o...

For the normal linear model variable selection problem, we propose selection criteria based on a fully Bayes formulation with a generalization of Zellner's g-prior which allows for p &gt; n. A special case of the prior formulation is seen to yield tractable closed forms for marginal densities and Bayes factors which reveal new model evaluation characteristics of potential interest.

We derive minimax generalized Bayes estimators of regression coefficients in the general linear model with spherically symmetric errors under invariant quadratic loss for the case of unknown scale. The class of estimators generalizes the class considered in Maruyama and Strawderman [Y. Maruyama, W.E. Strawderman, A new class of generalized Bayes minimax ridge regression estimators, Ann. Statist., 33 (2005) 1753-1770] to include non-monotone shrinkage functions. (C) 2009 Elsevier Inc. All rights reserved.

A sufficient condition for the admissibility of generalized Bayes estimators of the location vector of spherically symmetric distribution under squared error loss is derived. This is as strong a condition as that of Brown [L. D. Brown, Admissible estimators, recurrent diffusions, and insoluble boundary value problems, Ann. Math. Statist. 42 (1971) 855-903] undernormality. In particular we establish the admissibility of generalized Bayes estimators with respect to the harmonic prior and priors with slightly heavier tails than the harmonic prior. The key to our proof is an adaptive sequence of smooth proper priors approaching an improper prior fast enough to establish admissibility. (C) 2009 Elsevier Inc. All rights reserved.

We give a sufficient condition for admissibility of generalized Bayes estimators of the location vector of spherically symmetric distribution under squared error loss. Compared to the known results for the multivariate normal case, our sufficient condition is very tight and is close to being a necessary condition. In particular, we establish the admissibility of generalized Bayes estimators with respect to the harmonic prior and priors with slightly heavier tail than the harmonic prior. We use the theory of regularly varying functions to construct a sequence of smooth proper priors approaching an improper prior fast enough for establishing the admissibility. We also discuss conditions of minimaxity of the generalized Bayes estimator with respect to the harmonic prior. © 2007 Elsevier Inc. All rights reserved.

地球統計学において,観測された情報(観測値とその位置)に基づいて,未観測地点での値を予測する問題を考える.標準的には,最良線形不偏予測量が用いられるが,本論文では縮小型のベイズ予測量が,平均二乗誤差の意味で最良不偏予測量よりも良いことを示す.

We consider estimation of a multivariate normal mean vector under sum of<br /> squared error loss. We propose a new class of smooth estimators parameterized<br /> by \alpha dominating the James-Stein estimator. The estimator for \alpha=1<br /> corresponds to the generalized Bayes estimator with respect to the harmonic<br /> prior. When \alpha goes to infinity, the estimator converges to the James-Stein<br /> positive-part estimator. Thus the class of our estimators is a bridge between<br /> the admissible estimator (\alpha=1) and the inadmissible estimator<br /> (\alpha=\infty). Although the estimators have quasi-admissibility whic...

A new class of minimax generalized Bayes estimators of the variance of a normal distribution is given under both quadratic and entropy losses. One contribution of the paper is a new class of minimax generalized Bayes estimators of a particularly simple form. Another contribution is a class of minimax generalized Bayes procedures satisfying a Strawderman [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190-198]-type condition which do not satisfy a Brewster and Zidek [1974. Improving on equivariant estimators. Ann. Statist. 2, 21-38]-type condition. We indicate that the new class may have a noticeably larger region of substantial improvement over the usual estimator than Brewster and Zidek-type procedures. (c) 2005 Elsevier B.V. All rights reserved.

Let y = A beta + epsilon, where y is an N x 1 vector of observations, beta is a p x I vector of unknown regression coefficients, A is an N x p design matrix and E is a spherically symmetric error term with unknown scale parameter a. We consider estimation of under general quadratic loss functions, and, in particular, extend the work of Strawderman [J. Amer Statist. Assoc. 73 (1978) 623-627] and Casella [Ann. Statist. 8 (1980) 1036-1056, J. Amer. Statist. Assoc. 80 (1985) 753-758] by finding adaptive minimax estimators (which are, under the nonnality assumption, also generalized Bayes) of beta, which have greater numerical stability (i.e., smaller condition number) than the usual least squares estimator. In particular, we give a subclass of such estimators which, surprisingly, has a very simple form. We also show that under certain conditions the generalized Bayes minimax estimators in the normal case are also generalized Bayes and minimax in the general case of spherically symmetric errors.

This paper develops necessary conditions for an estimator to dominate the James-Stein estimator and hence the James-Stein positive-part estimator. The ultimate goal is to find classes of such dominating estimators which are admissible. While there are a number of results giving classes of estimators dominating the James-Stein estimator, the only admissible estimator known to dominate the James-Stein estimator is the generalized Bayes estimator relative to the fundamental harmonic function in three and higher dimension. The prior was suggested by Stein and the domination result is due to Kubokawa. Shao and Strawderman gave a class of estimators dominating the James-Stein positive-part estimator but were unable to demonstrate admissibility of any in their class. Maruyama, following a suggestion of Stein, has studied generalized Bayes estimators which are members of a point mass at zero and a prior similar to the harmonic prior. He finds a subclass which is minimax and admissible but is unable to show that any in his class with positive point mass at zero dominate the James-Stein estimator. The results in this paper show that a subclass of Maruyama's procedures including the class that Stein conjectured might contain members dominating the James-Stein estimator cannot dominate the James-Stein estimator. We also show that under reasonable conditions, the "constant" in shrinkage factor must approach p - 2 for domination to hold.

On the problem of estimating a positive normal mean with known variance, it is well known that one minimax admissible estimator is the generalized Bayes one with respect to the non-informative prior measure, the Lebesgue measure, restricted on the positive half-line. When the true variance is misspecified, however, it is shown that this estimator does not always retain minimaxity and admissibility. In particular, it is almost surely inadmissible in the misspecification case.

We consider estimation of a multivariate normal mean vector under sum of squared error loss. We propose a new class of minimax admissible estimator which are generalized Bayes with respect to a prior distribution which is a mixture of a point prior at the origin and a continuous hierarchical type prior. We also study conditions under which these generalized Bayes minimax estimators improve on the James-Stein estimator and on the positive-part James-Stein estimator. (C) 2003 Elsevier Inc. All rights reserved.

The problem of estimating a mean vector of scale mixtures of multivariate normal distributions with the quadratic loss function is considered. For a certain class of these distributions, which includes at least multivariate-t distributions, admissible minimax estimators are given. (C) 2003 Elsevier Science (USA). All rights reserved.

In the estimation of a multivariate normal mean for the case where the unknown covariance matrix is proportional to the identity matrix, A class of generalized Bayes estimators dominating the James-Stein rule is obtained. It is noted that a sequence of estimators in our class converges to the positive-part James-Stein estimator. © 2014, Oldenbourg Wissenschaftsverlag GmbH, Rosenheimer Str. 145, 81671 München. All rights reserved.

The problem of estimating the mean of a multivariate normal distribution is considered. A class of admissible minimax estimators is constructed. This class includes two well-known classes of estimators, Strawderman's and Alam's. Further, this class is much broader than theirs. (C) 1998 Academic Press.

In the estimation problem of unknown variance of a multivariate normal distribution, a new class of minimax estimators is obtained. It is noted that a sequence of estimators in our class converges to the Stein's truncated estimator.

The problem of estimating the quadratic loss function for the estimator of a multivariate normal mean is considered. A positive estimator which dominates Johnstone (1987)'s shrinkage rule is given. (C) 1997 Published by Elsevier Science B.V.