Stein Phenomenon Research Articles

Sharp Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane

The main purpose of this paper is to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane using the Helgason–Fourier analysis on symmetric spaces. A crucial part of our work is to establish appropriate factorization theorems on these spaces, which can be of independent interest. To this end, we need to identify and introduce the “quaternionic Geller operators” and the “octonionic Geller operators”, which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason–Fourier analysis on symmetric spaces, some precise estimates for the heat and the Bessel–Green– Riesz kernels, and the Kunze–Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient to establish the Adams and Hardy–Adams inequalities on these spaces. This paper, together with our earlier works on real and complex hyperbolic spaces, completes our study of the factorization theorems, higher order Poincaré–Sobolev, Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on all rank one symmetric spaces of noncompact type.

Estimates for matrix coefficients of representations

Estimates for matrix coefficients of unitary representations of semisimple Lie groups have been studied for a long time, starting with the seminal work by Bargmann, by Ehrenpreis and Mautner, and by Kunze and Stein. Two types of estimates have been established: on the one hand, $L^p$ estimates, which are a dual formulation of the Kunze--Stein phenomenon, and which hold for all matrix coefficients, and on the other pointwise estimates related to asymptotic expansions at infinity, which are more precise but only hold for a restricted class of matrix coefficients. In this paper we prove a new type of estimate for the irreducibile unitary representations of ${\rm SL}(2,\Bbb{R})$ and for the so-called metaplectic representation, which we believe has the best features of, and implies, both forms of estimate described above. As an application outside representation theory, we prove a new $L^2$ estimate of dispersive type for the free Schr\"odinger equation in $\Bbb{R}^n$.

Density Prediction and the Stein Phenomenon

The Stein phenomenon is a path-breaking discovery in mathematical statistics in the last century. A large number of researchers followed Stein’s footsteps and developed a wide variety of minimax shrinkage point estimators of a multivariate normal mean vector, each dominating the sample mean. More recently, the problem resurfaced, but this time with minimax shrinkage predictive density estimation, illustrating once again the Stein phenomenon In this review paper, we discuss parallel developments for normal and Poisson distributions under the Kullback-Leibler and more general divergence losses.

An endpoint estimate of the Kunze–Stein phenomenon on SU(n,m)

An endpoint estimate of the Kunze–Stein phenomenon on SU(n,m)

The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth

Let X be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of non-compact harmonic manifolds except for the flat spaces. Denote by $$h > 0$$ the mean curvature of horospheres in X, and set $$\rho = h/2$$ . Fixing a basepoint $$o \in X$$ , for $$\xi \in \partial X$$ , denote by $$B_{\xi }$$ the Busemann function at $$\xi $$ such that $$B_{\xi }(o) = 0$$ . Then for $$\lambda \in \mathbb {C}$$ the function $$e^{(i\lambda - \rho )B_{\xi }}$$ is an eigenfunction of the Laplace–Beltrami operator with eigenvalue $$-(\lambda ^2 + \rho ^2)$$ . For a function f on X, we define the Fourier transform of f by $$\begin{aligned} \tilde{f}(\lambda , \xi ) := \int _X f(x) e^{(-i\lambda -\rho )B_{\xi }(x)} \mathrm{{d}}vol(x) \end{aligned}$$ for all $$\lambda \in \mathbb {C}, \xi \in \partial X$$ for which the integral converges. We prove a Fourier inversion formula $$\begin{aligned} f(x) = C_0 \int _{0}^{\infty } \int _{\partial X} \tilde{f}(\lambda , \xi ) e^{(i\lambda - \rho )B_{\xi }(x)} \mathrm{{d}}\lambda _o(\xi ) |c(\lambda )|^{-2} \mathrm{{d}}\lambda \end{aligned}$$ for $$f \in C^{\infty }_c(X)$$ , where c is a certain function on $$\mathbb {R} - \{0\}$$ , $$\lambda _o$$ is the visibility measure on $$\partial X$$ with respect to the basepoint $$o \in X$$ and $$C_0 > 0$$ is a constant. We also prove a Plancherel theorem, and a version of the Kunze–Stein phenomenon.

Data-Pooling in Stochastic Optimization

Managing large-scale systems often involves simultaneously solving thousands of unrelated stochastic optimization problems, each with limited data. Intuition suggests one can decouple these unrelated problems and solve them separately without loss of generality. We propose a novel data-pooling algorithm called Shrunken-SAA that disproves this intuition. In particular, we prove that combining data across problems can outperform decoupling, even when there is no a priori structure linking the problems and data are drawn independently. Our approach does not require strong distributional assumptions and applies to constrained, possibly non-convex, non-smooth optimization problems such as vehicle-routing, economic lot-sizing or facility location. We compare and contrast our results to a similar phenomenon in statistics (Stein's Phenomenon), highlighting unique features that arise in the optimization setting that are not present in estimation. We further prove that as the number of problems grows large, Shrunken-SAA learns if pooling can improve upon decoupling and the optimal amount to pool, even if the average amount of data per problem is fixed and bounded. Importantly, we highlight a simple intuition based on stability that highlights when and why data-pooling offers a benefit, elucidating this perhaps surprising phenomenon. This intuition further suggests that data-pooling offers the most benefits when there are many problems, each of which has a small amount of relevant data. Finally, we demonstrate the practical benefits of data-pooling using real data from a chain of retail drug stores in the context of inventory management.

Operational calculus and integral transforms for groups with finite propagation speed

AbstractLet

Decision-theoretic issues in heterogeneity variance estimation

Motivated by a heteroscedastic random effects setting of meta-analysis, a general model for the between-study variance is studied from the decision-theoretic point of view. This model leads to estimation of a linear in the variance reciprocals, random function or to simultaneous inference on curve-confined natural parameters of independent heterogeneous \(\chi ^2\)-random variables with given degrees of freedom. A form of the Stein phenomenon for the suggested loss functions is noted; the exact minimax value is determined, and minimax estimators are derived.

Wiener–Tauberian type theorems for radial sections of homogenous vector bundles on certain rank one Riemannian symmetric spaces of noncompact type

We will show that an uniform treatment yields Wiener–Tauberian type results for various Banach algebras and modules consisting of radial sections of some homogenous vector bundles on rank one Riemannian symmetric spaces G/K of noncompact type. One example of such a vector bundle is the spinor bundle. The algebras and modules we consider are natural generalizations of the commutative Banach algebra of integrable radial functions on G/K. The first set of them are Beurling algebras with analytic weights, while the second set arises due to Kunze–Stein phenomenon for noncompact semisimple Lie groups.

The Stein phenomenon for monotone incomplete multivariate normal data

We establish the Stein phenomenon in the context of two-step, monotone incomplete data drawn from N p + q ( μ , Σ ) , a ( p + q ) -dimensional multivariate normal population with mean μ and covariance matrix Σ . On the basis of data consisting of n observations on all p + q characteristics and an additional N − n observations on the last q characteristics, where all observations are mutually independent, denote by μ ̂ the maximum likelihood estimator of μ . We establish criteria which imply that shrinkage estimators of James–Stein type have lower risk than μ ̂ under Euclidean quadratic loss. Further, we show that the corresponding positive-part estimators have lower risk than their unrestricted counterparts, thereby rendering the latter estimators inadmissible. We derive results for the case in which Σ is block-diagonal, the loss function is quadratic and non-spherical, and the shrinkage estimator is constructed by means of a nondecreasing, differentiable function of a quadratic form in μ ̂ . For the problem of shrinking μ ̂ to a vector whose components have a common value constructed from the data, we derive improved shrinkage estimators and again determine conditions under which the positive-part analogs have lower risk than their unrestricted counterparts.

On the Stein phenomenon under divergence loss and an unknown variance–covariance matrix

The paper develops a general class of shrinkage estimators for estimating the normal mean, which dominates the sample mean in three or higher dimensions under a general divergence loss. In the process, the earlier works of James and Stein [11] and Efron and Morris [5] are generalized considerably.

Holomorphic [formula omitted]-type for sub-Laplacians on connected Lie groups

We study the problem of determining all connected Lie groups G which have the following property (hlp): every sub-Laplacian L on G is of holomorphic L p -type for 1 ⩽ p < ∞ , p ≠ 2 . First we show that semi-simple non-compact Lie groups with finite center have this property, by using holomorphic families of representations in the class one principal series of G and the Kunze–Stein phenomenon. We then apply an L p -transference principle, essentially due to Anker, to show that every connected Lie group G whose semi-simple quotient by its radical is non-compact has property (hlp). For the convenience of the reader, we give a self-contained proof of this transference principle, which generalizes the well-known Coifman–Weiss principle. One is thus reduced to studying those groups for which the semi-simple quotient is compact, i.e. to compact extensions of solvable Lie groups. In this article, we consider semi-direct extensions of exponential solvable Lie groups by connected compact Lie groups. It had been proved in joint work by W. Hebisch and the two first named authors that every exponential solvable Lie group S, which has a non-∗ regular co-adjoint orbit whose restriction to the nilradical is closed, has property (hlp), and we show here that (hlp) remains valid for compact extensions of these groups.

Estimation, prediction and the Stein phenomenon under divergence loss

We consider two problems: (1) estimate a normal mean under a general divergence loss introduced in [S. Amari, Differential geometry of curved exponential families — curvatures and information loss, Ann. Statist. 10 (1982) 357–387] and [N. Cressie, T.R.C. Read, Multinomial goodness-of-fit tests, J. Roy. Statist. Soc. Ser. B. 46 (1984) 440–464] and (2) find a predictive density of a new observation drawn independently of observations sampled from a normal distribution with the same mean but possibly with a different variance under the same loss. The general divergence loss includes as special cases both the Kullback–Leibler and Bhattacharyya–Hellinger losses. The sample mean, which is a Bayes estimator of the population mean under this loss and the improper uniform prior, is shown to be minimax in any arbitrary dimension. A counterpart of this result for predictive density is also proved in any arbitrary dimension. The admissibility of these rules holds in one dimension, and we conjecture that the result is true in two dimensions as well. However, the general Baranchick [A.J. Baranchick, a family of minimax estimators of the mean of a multivariate normal distribution, Ann. Math. Statist. 41 (1970) 642–645] class of estimators, which includes the James–Stein estimator and the Strawderman [W.E. Strawderman, Proper Bayes minimax estimators of the multivariate normal mean, Ann. Math. Statist. 42 (1971) 385–388] class of estimators, dominates the sample mean in three or higher dimensions for the estimation problem. An analogous class of predictive densities is defined and any member of this class is shown to dominate the predictive density corresponding to a uniform prior in three or higher dimensions. For the prediction problem, in the special case of Kullback–Leibler loss, our results complement to a certain extent some of the recent important work of Komaki [F. Komaki, A shrinkage predictive distribution for multivariate normal observations, Biometrika 88 (2001) 859–864] and George, Liang and Xu [E.I. George, F. Liang, X. Xu, Improved minimax predictive densities under Kullbak–Leibler loss, Ann. Statist. 34 (2006) 78–92]. While our proposed approach produces a general class of predictive densities (not necessarily Bayes, but not excluding Bayes predictors) dominating the predictive density under a uniform prior. We show also that various modifications of the James–Stein estimator continue to dominate the sample mean, and by the duality of estimation and predictive density results which we will show, similar results continue to hold for the prediction problem as well.

Stein's phenomenon in estimation of means restricted to a polyhedral convex cone

This paper treats the problem of estimating the restricted means of normal distributions with a known variance, where the means are restricted to a polyhedral convex cone which includes various restrictions such as positive orthant, simple order, tree order and umbrella order restrictions. In the context of the simultaneous estimation of the restricted means, it is of great interest to investigate decision-theoretic properties of the generalized Bayes estimator against the uniform prior distribution over the polyhedral convex cone. In this paper, the generalized Bayes estimator is shown to be minimax. It is also proved that it is admissible in the one- or two-dimensional case, but is improved on by a shrinkage estimator in the three- or more-dimensional case. This means that the so-called Stein phenomenon on the minimax generalized Bayes estimator can be extended to the case where the means are restricted to the polyhedral convex cone. The risk behaviors of the estimators are investigated through Monte Carlo simulation, and it is revealed that the shrinkage estimator has a substantial risk reduction.

The Kunze–Stein phenomenon on the isometry group of a tree

Let G be the group of isometries of a homogeneous tree . In the first part of this paper we decompose G in terms of certain subgroups N, ℤ, and K to obtain the related integral formula Then, by using ideas of A. Ionescu and the formula above, we prove that and that a related maximal operator on  is bounded from L2, 1() to L2,∞(). We finally show that Lp, 1(K\G/K) is a commutative Banach algebra of convolutors for Lp(G) and give an explicit description of its Gelfand spectrum.

On the concepts of admissibility and coherence

In the present paper it will be argued that if a parameter value assigns zero probability to an open set containing the actual response, then that parameter value should be excluded from the parameter space. When this is done, the model becomes a restricted model, and sampling theory inferences should be focused on the sampling distribution of an hypothetical independent response from the restricted model. The above situation may arise when probability densities vanish outside a compact set. This phenomenon arises frequently in the real world, but it is usually ignored for reasons of mathematical convenience. Although many statistical procedures remain substantially the same when we consider this restricted model, admissibility propertics may be drastically changed, and inferences which are known to be inadmissible may turn out to be really admissible. Thus, Stein's phenomenon concerning the inadmissibility of the sample mean as an estimator of the population mean of ap-variate normal distribution whenp≥3may be explained by the fact that the distribution has a compact support but this has been ignored by reasons of mathematical convenience. The introduction of a restricted model is also important in the study of coherence. Thus, it will be shown that Brunk's theory of countably additive coherence, which admits the use of countably additive improper priors, can be improved with the introduction of a restricted model. Thus, the theory will be unified because it will be proved that the posterior is really coherent if and only if it is a Bayes posterior, and it will be simplified because it will not be required that the prior be minimally compatible with the model.

Some limitation on stein's phenomenon

We show that Stein's phenomenon is impossible for the following decision setting: the parameter space is compact, the loss function is the sum of coordinatewise strictly convex continuous loss functions, and the probability density function having the same support for every parameter θ is continuous in θ. Therefore, any coordinatewise admissible decision rule is admissible.

Estimating the mean function of a Gaussian process and the Stein effect

The problem of global estimation of the mean function θ(·) of a quite arbitrary Gaussian process is considered. The loss function in estimating θ by a function a(·) is assumed to be of the form L( θ, a) = ∫ [ θ( t) − a( t)] 2 μ( dt), and estimators are evaluated in terms of their risk function (expected loss). The usual minimax estimator of θ is shown to be inadmissible via the Stein phenomenon; in estimating the function θ we are trying to simultaneously estimate a larger number of normal means. Estimators improving upon the usual minimax estimator are constructed, including an estimator which allows the incorporation of prior information about θ. The analysis is carried out by using a version of the Karhunen-Loéve expansion to represent the original problem as the problem of estimating a countably infinite sequence of means from independent normal distributions.