two level linear programming decentralized planning has long been recognized as an important decision making problem many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which often exist in practice multilevel programming models partition control over decision variables among ordered levels within a hierarchical planning structure a planner at one level of the hierarchy may have his objective function and set of feasible decisions determined in part by other levels however his control instruments may allow him to influence the policies at other levels and thereby improve his own objective function this paper examines the special case of the two level linear programming problem geometric characterizations and algorithms are presented with some examples the goal is to demonstrate the tractability of such problems and motivate a wider interest in their study

learning multiple tasks with kernel methods we study the problem of learning many related tasks simultaneously using kernel methods and regularization the standard single task kernel methods such as support vector machines and regularization networks are extended to the case of multi task learning our analysis shows that the problem of estimating many task functions with regularization can be cast as a single task learning problem if a family of multi task kernel functions we define is used these kernels model relations among the tasks and are derived from a novel form of regularizers specific kernels that can be used for multi task learning are provided and experimentally tested on two real data sets in agreement with past empirical work on multi task learning the experiments show that learning multiple related tasks simultaneously using the proposed approach can significantly outperform standard single task learning particularly when there are many related tasks but few data per task

dimension reduction and coefficient estimation in multivariate linear regression summary we introduce a general formulation for dimension reduction and coefficient estimation in the multivariate linear model we argue that many of the existing methods that are commonly used in practice can be formulated in this framework and have various restrictions we continue to propose a new method that is more flexible and more generally applicable the method proposed can be formulated as a novel penalized least squares estimate the penalty that we employ is the coefficient matrix s ky fan norm such a penalty encourages the sparsity among singular values and at the same time gives shrinkage coefficient estimates and thus conducts dimension reduction and coefficient estimation simultaneously in the multivariate linear model we also propose a generalized cross validation type of criterion for the selection of the tuning parameter in the penalized least squares simulations and an application in financial econometrics demonstrate competitive performance of the new method an extension to the non parametric factor model is also discussed

joint continuum regression for multiple predictands abstract this article generalizes continuum regression cr in the hope that regressors jointly constructed for several predictands might improve on the separate prediction of individual predictands the generalization developed is a mixture of principal components regression and de jong s modification of partial least squares for multiple predictands the balance of ingredients can be chosen by cross validation as can the number of regressors constructed the new method has been tested on real and simulated data the indications are that conditions for the superiority of the joint approach may be rare in practice

multiple output regression with latent noise in high dimensional data structured noise caused by observed and unobserved factors affecting multiple target variables simultaneously imposes a serious challenge for modeling by masking the often weak signal therefore 1 explaining away the structured noise in multiple output regression is of paramount importance additionally 2 assumptions about the correlation structure of the regression weights are needed we note that both can be formulated in a natural way in a latent variable model in which both the interesting signal and the noise are mediated through the same latent factors under this assumption the signal model then borrows strength from the noise model by encouraging similar effects on correlated targets we introduce a hyperparameter for the latent signal to noise ratio which turns out to be important for modelling weak signals and an ordered infinite dimensional shrinkage prior that resolves the rotational unidentifiability in reduced rank regression models simulations and prediction experiments with metabolite gene expression fmri measurement and macroeconomic time series data show that our model equals or exceeds the state of the art performance and in particular outperforms the standard approach of assuming independent noise and signal models

flexible discriminant analysis by optimal scoring abstract fisher s linear discriminant analysis is a valuable tool for multigroup classification with a large number of predictors one can find a reduced number of discriminant coordinate functions that are optimal for separating the groups with two such functions one can produce a classification map that partitions the reduced space into regions that are identified with group membership and the decision boundaries are linear this article is about richer nonlinear classification schemes linear discriminant analysis is equivalent to multiresponse linear regression using optimal scorings to represent the groups in this paper we obtain nonparametric versions of discriminant analysis by replacing linear regression by any nonparametric regression method in this way any multiresponse regression technique such as mars or neural networks can be postprocessed to improve its classification performance

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

fast active appearance model search using canonical correlation analysis a fast aam search algorithm based on canonical correlation analysis cca aam is introduced it efficiently models the dependency between texture residuals and model parameters during search experiments show that cca aams while requiring similar implementation effort consistently outperform standard search with regard to convergence speed by a factor of four

an interpretation of partial least squares abstract univariate partial least squares pls is a method of modeling relationships between a y variable and other explanatory variables it may be used with any number of explanatory variables even far more than the number of observations a simple interpretation is given that shows the method to be a straightforward and reasonable way of forming prediction equations its relationship to multivariate pls in which there are two or more y variables is examined and an example is given in which it is compared by simulation with other methods of forming prediction equations with univariate pls linear combinations of the explanatory variables are formed sequentially and related to y by ordinary least squares regression it is shown that these linear combinations here called components may be viewed as weighted averages of predictors where each predictor holds the residual information in an explanatory variable that is not contained in earlier components and the quantity to be predicted is the vecto

singular value shrinkage priors for bayesian prediction we develop singular value shrinkage priors for the mean matrix parameters in the matrix variate normal model with known covariance matrices introduced priors are superharmonic and put more weight on matrices with smaller singular values they are a natural generalization of the stein prior bayes estimators and bayesian predictive densities based on introduced priors are minimax and dominate those based on the uniform prior in finite samples the risk reduction is large when the true value of the parameter has small singular values in particular introduced priors perform stably well in the low rank case we apply this result to multivariate linear regression problems by considering priors depending on the future samples introduced priors are compatible with reduced rank regression

hierarchical optimization a satisfactory solution abstract hierarchical optimization or multi level programming techniques are extensions of stackelberg games for solving decentralized planning problems with multiple decision makers in a hierarchical organization they become more important for contemporary decentralized organizations where each unit or department seeks its own interests traditional approaches include vertex enumeration algorithms and approaches based on kuhn tucker conditions or penalty functions these are not only technically inefficient but also lead to a paradox that the follower s decision power dominates the leader s in this study concepts of memberships of optimalities and degrees of decision powers are proposed to solve the above problems efficiently in the proposed hierarchy decision process the leader first sets memberships of optimalities of his her possible objective values and decisions as well as his her decision power and then asks the follower for his her optima calculated in isolation under given constraints the follower s decision with the corresponding levels of optimalities and decision powers are submitted to and modified by the leader with considerations of overall benefit for the organization and distribution of decision power until a best preferred solution is reached

random calibration with many measurements an application of stein estimation in the problem considered a vector of many imprecise measurements e g spectroscopic is used to linearly predict a quantity whose precise measurement is difficult or expensive the regression vector is estimated from a calibration experiment having both types of measurements for a random sample most previous approaches to this problem adjust for approximate multicollinearity which often results from the correlations among the imprecise measurements by inverting an approximation e g factor analytic to their covariance matrix in the approach here it is argued that the regression vector should lie in a lower dimensional suhspace determined by the principal components of the covariance matrix it is then estimated by applying a stein contraction of the least squares estimator to the principal components regression estimator examples using real data are presented in which the proposed estimator substantially improves on the ordinary least squares estimation

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

predicting multivariate responses in multiple linear regression we look at the problem of predicting several response variables from the same set of explanatory variables the question is how to take advantage of correlations between the response variables to improve predictive accuracy compared with the usual procedure of doing individual regressions of each response variable on the common set of predictor variables a new procedure is introduced called the curds and whey method its use can substantially reduce prediction errors when there are correlations between responses while maintaining accuracy even if the responses are uncorrelated in extensive simulations the new procedure is compared with several previously proposed methods for predicting multiple responses including partial least squares and exhibits superior accuracy one version can be easily implemented in the context of standard statistical packages

regularized partial and or constrained redundancy analysis methods of incorporating a ridge type of regularization into partial redundancy analysis pra constrained redundancy analysis cra and partial and constrained redundancy analysis pcra were discussed the usefulness of ridge estimation in reducing mean square error mse has been recognized in multiple regression analysis for some time especially when predictor variables are nearly collinear and the ordinary least squares estimator is poorly determined the ridge estimation method was extended to pra cra and pcra where the reduced rank ridge estimates of regression coefficients were obtained by minimizing the ridge least squares criterion it was shown that in all cases they could be obtained in closed form for a fixed value of ridge parameter an optimal value of the ridge parameter is found by g fold cross validation illustrative examples were given to demonstrate the usefulness of the method in practical data analysis situations

an efficient point algorithm for a linear two stage optimization problem this paper presents an algorithm using sensitivity analysis to solve a linear two stage optimization problem the underlying theory rests on a set of first order optimality conditions that parallel the kuhn tucker conditions associated with a one dimensional parametric linear program the solution to the original problem is uncovered by systematically varying the parameter over the unit interval and solving the corresponding linear program finite convergence is established under nondegenerate assumptions the paper also discusses other solution techniques including branch and bound and vertex enumeration and gives an example highlighting their computational and storage requirements by these measures the algorithm presented here has an overall advantage finally a comparison is drawn between bicriteria and bilevel programming and underscored by way of an example

bayes model averaging with selection of regressors when a number of distinct models contend for use in prediction the choice of a single model can offer rather unstable predictions in regression stochastic search variable selection with bayesian model averaging offers a cure for this robustness issue but at the expense of requiring very many predictors here we look at bayes model averaging incorporating variable selection for prediction this offers similar mean square errors of prediction but with a vastly reduced predictor space this can greatly aid the interpretation of the model it also reduces the cost if measured variables have costs the development here uses decision theory in the context of the multivariate general linear model in passing this reduced predictor space bayes model averaging is contrasted with single model approximations a fast algorithm for updating regressions in the markov chain monte carlo searches for posterior inference is developed allowing many more variables than observations to be contemplated we discuss the merits of absolute rather than proportionate shrinkage in regression especially when there are more variables than observations the methodology is illustrated on a set of spectroscopic data used for measuring the amounts of different sugars in an aqueous solution

near optimal coresets for least squares regression we study the constrained least squares regression as well as multiple response least squares regression and ask the question of whether a subset of the data a coreset suffices to compute a good approximate solution to the regression we give deterministic low order polynomial time algorithms to construct such coresets with approximation guarantees together with lower bounds indicating that there is not much room for improvement upon our results

interactive fuzzy programming for multilevel linear programming problems this paper presents interactive fuzzy programming for multilevel linear programming problems in fuzzy programming for multilevel linear programming problems recently developed by lai et al since the fuzzy goals are determined for both an objective function and decision variables at the upper level undesirable solutions are produced when these fuzzy goals are inconsistent in order to overcome such problems after eliminating the fuzzy goals for decision variables interactive fuzzy programming for multilevel linear programming problems is presented in our interactive method after determining the fuzzy goals of the decision makers at all levels a satisfactory solution is derived efficiently by updating the satisfactory degrees of decision makers at the upper level with considerations of overall satisfactory balance among all levels illustrative numerical examples for two level and for three level linear programming problems are provided to demonstrate the feasibility of the proposed method

computational difficulties of bilevel linear programming we show using small examples that two algorithms previously published for the bilevel linear programming problem blp may fail to find the optimal solution and thus must be considered to be heuristics a proof is given that solving blp problems is np hard which makes it unlikely that there is a good exact algorithm

continuum regression and ridge regression we demonstrate the close relationship between first factor continuum regression and standard ridge regression the difference is that continuum regression inserts a scalar compensation factor for that part of the shrinkage in ridge regression that has no connection with tendencies towards collinearity we interpret this to mean that first factor continuum regression is preferable in principle to ridge regression if we want protection against near colliearity but do not admit shrinkage as a general principle furthermore our experience indicates that with first factor continuum regression we can obtain predictors that are at least as mean squared error efficient as with ridge regression but with less sensitivity to the choice of ridge constant

ep gig priors and applications in bayesian sparse learning in this paper we propose a novel framework for the construction of sparsity inducing priors in particular we define such priors as a mixture of exponential power distributions with a generalized inverse gaussian density ep gig ep gig is a variant of generalized hyperbolic distributions and the special cases include gaussian scale mixtures and laplace scale mixtures furthermore laplace scale mixtures can subserve a bayesian framework for sparse learning with nonconvex penalization the densities of ep gig can be explicitly expressed moreover the corresponding posterior distribution also follows a generalized inverse gaussian distribution we exploit these properties to develop em algorithms for sparse empirical bayesian learning we also show that these algorithms bear an interesting resemblance to iteratively reweighted l2 or l1 methods finally we present two extensions for grouped variable selection and logistic regression

robust multivariate regression we introduce a robust method for multivariate regression based on robust estimation of the joint location and scatter matrix of the explanatory and response variables as a robust estimator of location and scatter we use the minimum covariance determinant mcd estimator of rousseeuw based on simulations we investigate the finite sample performance and robustness of the estimator to increase the efficiency we propose a reweighted estimator selected from several possible reweighting schemes the resulting multivariate regression does not need much computation time and is applied to real datasets we show that the multivariate regression estimator has the appropriate equivariance properties has a bounded influence function and inherits the breakdown value of the mcd estimator these theoretical robustness properties confirm the good finite sample results obtained from the simulations

l2rm low rank linear regression models for high dimensional matrix responses the aim of this article is to develop a low rank linear regression model to correlate a high dimensional response matrix with a high dimensional vector of covariates when coefficient matrices have

penalized normal likelihood and ridge regularization of correlation and covariance matrices high dimensionality causes problems in various areas of statistics a particular situation that rarely has been considered is the testing of hypotheses about multivariate regression models in which

nonsingularity under data rounding abstract for a square floating point matrix a we give a test for the existence of a singular real matrix which rounds to a

comparison of prediction methods when only a few components are relevant abstract we consider prediction in a multiple regression model where we also look on the explanatory variables as random if the number of explanatory variables is large then the common least squares multiple regression solution may not be the best one we give a methodology for comparing certain alternative prediction methods by asymptotic calculations and perform such a comparisons for four specific methods the results indicate that none of these methods dominates the others and that the difference between the methods typically but not always is small when the number of observations is large in particular principal component regression does well when the eigenvalues corresponding to components not correlated with the dependent variables i e the irrelevant eigenvalues are extremely small or extremely large partial least squares regression does well for intermediate irrelevant eigenvalues a maximum likelihood type method dominates the others asymptotically at least in the case of one relevan

efficient solutions for the linear bilevel programming problem abstract one of the interesting features of the bilevel programming problem is that its optimal solution may not be pareto optimal this behavior may be economically inadmissible this paper will present a post optimality analysis it is intended to provide some rational efficient solutions for decision makers after a non pareto optimal solution is obtained for a linear bilevel programming problem an efficient solution which obeys some properties will be adopted by decision makers according to their decision preferences

automatic tagging with existing and novel tags summaryautomatic tagging by key words and phrases is important in multi label classification of a document in this paper we first introduce a tagging loss to measure the discrepancy between predicted and actual tag sets which is expressed in terms of a sum of weighted pairwise margins between two tags by their degree of similarity we then construct a regularized empirical loss to incorporate linguistic knowledge and identify a tagger maximizing the separations between the pairwise margins one salient feature of the proposed method is its capability to identify novel tags absent from a training sample by using their similarity to existing tags computationally the proposed method is implemented by an alternating direction method of multipliers integrated with a difference convex algorithm this permits scalable computation we show that the method achieves accurate tagging and that it compares favourably with existing methods finally we apply the proposed method to tagging a reuters news dataset

combining minimax shrinkage estimators abstract when one estimates a multivariate normal mean the use of stein estimation entails both the grouping of coordinates and the selection of a set of targets toward which to shrink each group in this article we propose new minimax multiple shrinkage estimators that allow for multiple specifications of these aspects we provide examples that are evaluated on real and simulated data including an estimator that adaptively resolves the issue of combining possibly related estimation problems and an adaptive clustering shrinkage estimator the construction and properties of these estimators are shown to follow from the application of the multiple shrinkage results of george 1986a to general partitioned shrinkage estimators

a brief survey of bandwidth selection for density estimation abstract there has been major progress in recent years in data based bandwidth selection for kernel density estimation some second generation methods including plug in and smoothed bootstrap techniques have been developed that are far superior to well known first generation methods such as rules of thumb least squares cross validation and biased cross validation we recommend a solve the equation plug in bandwidth selector as being most reliable in terms of overall performance this article is intended to provide easy accessibility to the main ideas for nonexperts

cross validation shrinkage of regression predictors summary copas 1983 suggests that shrinkage of predictors can be related to the least squares slope of actual on predicted values in a new set of data considering multiple regression in a cross validatory setting this shrinkage slope is approximated by an estimable function of the fitted residuals patterns of heteroscedasticity of residual variance can lead to more or less shrinkage of least squares a new predictor using a non parametric shrinkage multiplier is proposed

operator valued kernels for learning from functional response data in this paper we consider the problems of supervised classification and regression in the case where attributes and labels are functions a data is represented by a set of functions and the label is also a function we focus on the use of reproducing kernel hilbert space theory to learn from such functional data basic concepts and properties of kernel based learning are extended to include the estimation of function valued functions in this setting the representer theorem is restated a set of rigorously defined infinite dimensional operator valued kernels that can be valuably applied when the data are functions is described and a learning algorithm for nonlinear functional data analysis is introduced the methodology is illustrated through speech and audio signal processing experiments

canonical correlation analysis when the data are curves summary it is not immediately straightforward to extend canonical correlation analysis to the context of functional data analysis where the data are themselves curves or functions the obvious approach breaks down and it is necessary to use a method involving smoothing in some way such a method is introduced and discussed with reference to a data set on human gait the breakdown of the unsmoothed method is illustrated in a practical context and is demonstrated theoretically a consistency theorem for the smoothed method is proved in an increasing number of problems in a wide range of fields the data observed are not the univariate or multivariate observations of classical statistics but are func tions observed continuously ramsay and dalzell 1991 give the name functional data analysis to the analysis of data of this kind in most cases the observations will be functions of time or a closely related variable but there are clearly applications where the functions are surfaces observed over two or three dimensional space the motivating example for the present paper is a set of data on children s gait collected by the motion analysis laboratory at children s hospital san diego california see olshen et al 1989 for full details for each of a number of children several angles made by the child s joints knee hip etc are observed during the child s gait cycle one aim of the study is to gain understanding of the gait cycle of a normal child to make comparisons with children suffering from walking diffi culties these data motivated rice and silverman 1991 to discuss the extension of principal component analysis to the functional setting and to explain how smoothing can be incorporated into the analysis in a natural way in a substantial complemen tary paper ramsay and dalzell 1991 discuss different approaches to regression and principal component analysis in functional data analysis illustrating their work by a meteorological example the insights of functional data analysis may be helpful in chemometrics for instance a standard chemometric problem is the analysis of spectra observed in chromatography and these are to all intents and purposes functional observations another obvious area of potential relevance is the analysis of growth curve data the focus in the present paper will be on canonical correlation analysis cca in the gait example one might ask how variability in the knee angle cycle is related to

bayesian canonical correlation analysis canonical correlation analysis cca is a classical method for seeking correlations between two multivariate data sets during the last ten years it has received more and more attention in the machine learning community in the form of novel computational formulations and a plethora of applications we review recent developments in bayesian models and inference methods for cca which are attractive for their potential in hierarchical extensions and for coping with the combination of large dimensionalities and small sample sizes the existing methods have not been particularly successful in fulfilling the promise yet we introduce a novel efficient solution that imposes group wise sparsity to estimate the posterior of an extended model which not only extracts the statistical dependencies correlations between data sets but also decomposes the data into shared and data set specific components in statistics literature the model is known as inter battery factor analysis ibfa for which we now provide a bayesian treatment

minimax multiple shrinkage estimation on presente des nouveaux estimateurs qui incorporent l information partielle a priori en utilisant les donnees comme emulation du comportement et du risque des estimateurs consideres les plus efficaces

the solution and duality of imprecise network problems duality properties have been investigated by many researchers in the recent literature they are introduced in this paper for a fully fuzzified version of the minimal cost flow problem which is a basic model in network flow theory this model illustrates the least cost of the shipment of a commodity through a capacitated network in terms of the imprecisely known available supplies at certain nodes which should be transmitted to fulfil uncertain demands at other nodes first we review on the most valuable results on fuzzy duality concepts to facilitate the discussion of this paper by applying hukuhara s difference approximated and exact multiplication and wu s scalar production we exhibit the flow in network models then we use combinatorial algorithms on a reduced problem which is derived from fully fuzzified mcfp to acquire fuzzy optimal flows to give duality theorems we utilize a total order on fuzzy numbers due to the level of risk and realize optimality conditions for providing some efficient combinatorial algorithms finally we compare our results with the previous worthwhile works to demonstrate the efficiency and power of our scheme and the reasonability of our solutions in actual decision making problems

indirect multivariate response linear regression we propose a class of estimators of the multivariate response linear regression coefficient matrix that exploits the assumption that the response and predictors have a joint multivariate normal distribution this allows us to indirectly estimate the regression coefficient matrix through shrinkage estimation of the parameters of the inverse regression or the conditional distribution of the predictors given the responses we establish a convergence rate bound for estimators in our class and we study two examples which respectively assume that the inverse regression s coefficient matrix is sparse and rank deficient these estimators do not require that the forward regression coefficient matrix is sparse or has small frobenius norm using simulation studies we show that our estimators outperform competitors

simultaneous variable selection we propose a new method for selecting a common subset of explanatory variables where the aim is to model several response variables the idea is a natural extension of the lasso technique proposed by tibshirani 1996 and is based on the joint residual sum of squares while constraining the parameter estimates to lie within a suitable polyhedral region the properties of the resulting convex programming problem are analyzed for the special case of an orthonormal design for the general case we develop an efficient interior point algorithm the method is illustrated on a dataset with infrared spectrometry measurements on 14 qualitatively different but correlated responses using 770 wavelengths the aim is to select a subset of the wavelengths suitable for use as predictors for as many of the responses as possible

correlated logistic model with elastic net regularization for multilabel image classification in this paper we present correlated logistic corrlog model for multilabel image classification corrlog extends conventional logistic regression model into multilabel cases via explicitly modeling the pairwise correlation between labels in addition we propose to learn the model parameters of corrlog with elastic net regularization which helps exploit the sparsity in feature selection and label correlations and thus further boost the performance of multilabel classification corrlog can be efficiently learned though approximately by regularized maximum pseudo likelihood estimation and it enjoys a satisfying generalization bound that is independent of the number of labels corrlog performs competitively for multilabel image classification on benchmark data sets mulan scene mit outdoor scene pascal voc 2007 and pascal voc 2012 compared with the state of the art multilabel classification algorithms

on learning vector valued functions in this letter we provide a study of learning in a hilbert space of vectorvalued functions we motivate the need for extending learning theory of scalar valued functions by practical considerations and establish some basic results for learning vector valued functions that should prove useful in applications specifically we allow an output space y to be a hilbert space and we consider a reproducing kernel hilbert space of functions whose values lie in y in this setting we derive the form of the minimal norm interpolant to a finite set of data and apply it to study some regularization functionals that are important in learning theory we consider specific examples of such functionals corresponding to multiple output regularization networks and support vector machines for both regression and classification finally we provide classes of operator valued kernels of the dot product and translation invariant type

penalized discriminant analysis fisher s linear discriminant analysis lda is a popular data analytic tool for studying the relationship between a set of predictors and a categorical response in this paper we describe a penalized version of lda it is designed for situations in which there are many highly correlated predictors such as those obtained by discretizing a function or the grey scale values of the pixels in a series of images in cases such as these it is natural efficient and sometimes essential to impose a spatial smoothness constraint on the coefficients both for improved prediction performance and interpretability we cast the classification problem into a regression framework via optimal scoring using this our proposal facilitates the use of any penalized regression technique in the classification setting the technique is illustrated with examples in speech recognition and handwritten character recognition

how much does stein estimation help in multiple linear regression the use of stein estimation in multiple linear regression is considered tables and graphs are presented that compare the prediction mean squared errors of positive part james stein preliminary test reduced and full model least squares estimates the appropriateness of using stein contraction on possibly extraneous variables is emphasized and a procedure is presented for evaluating the likely savings in using stein estimation on the problem at hand an example is given

on the degrees of freedom of reduced rank estimators in multivariate regression we study the effective degrees of freedom of a general class of reduced rank estimators for multivariate regression in the framework of stein s unbiased risk estimation a finite sample exact unbiased estimator is derived that admits a closed form expression in terms of the thresholded singular values of the least squares solution and hence is readily computable the results continue to hold in the high dimensional setting where both the predictor and the response dimensions may be larger than the sample size the derived analytical form facilitates the investigation of theoretical properties and provides new insights into the empirical behaviour of the degrees of freedom in particular we examine the differences and connections between the proposed estimator and a commonly used naive estimator the use of the proposed estimator leads to efficient and accurate prediction risk estimation and model selection as demonstrated by simulation studies and a data example

estimation of a loss function for spherically symmetric distributions in the general linear model this paper is concerned with estimating the loss of a point estimator when sampling from a spherically symmetric distribution we examine the canonical setting of a general linear model where the dimension of the parameter space is greater than 4 and less than the dimension of the sampling space we consider two location estimators the least squares estimator and a shrinkage estimator and we compare their unbiased loss estimator with an improved loss estimator the domination results are valid for a large class of spherically symmetric distributions and in so far as the sampling distribution does not need to be precisely specified the estimates have desirable robustness properties