empirical bayes confidence intervals for selected parameters in high dimensional data modern statistical problems often involve a large number of populations and hence a large number of parameters that characterize these populations it is common for scientists to use data to select the most significant populations such as those with the largest t statistics the scientific interest often lies in studying and making inferences regarding these parameters called the selected parameters corresponding to the selected populations the current statistical practices either apply a traditional procedure assuming there were no selection a practice that is not valid or they use the bonferroni type procedure that is valid but very conservative and often noninformative in this article we propose valid and sharp confidence intervals that allow scientists to select parameters and to make inferences for the selected parameters based on the same data this type of confidence interval allows the users to zero in on the most interesting selected parameters without collecting more data the validity of

assessing the order of dependence for partially exchangeable binary data abstract the problem we consider is how to assess the order of serial dependence within partially exchangeable binary sequences we obtain exact conditional tests comparing any two orders by finding the conditional distribution of data given certain transition counts these tests are facilitated with a new monte carlo scheme asymptotic tests are also discussed in particular we show that the likelihood ratio tests have an asymptotic chi squared distribution thus generalizing the results of billingsley for the particular case of markov chains we apply these methods to several datasets and perform a simulation to study their properties

shrinkage estimators applied to prediction of french winter wheat yield summary the paper shows that it is possible to pool departmental information to predict regional wheat yield in an msep mean squared error of prediction context shrinkage estimation in linear agrometeorological models improves regional prediction cross validation is not precise enough to enable comparison between models the authors calculate the msep using the hypothesis of a multinormal distribution in simple cases the distribution parameters are estimated with data input and simulations are used for complex cases stein like estimators are shown to be a superior alternative to the conventional estimators usually applied for prediction

predictive updating methods with application to bayesian classification we propose algorithms based on random draws from predictive distributions of unknown quantities missing values for instance this procedure can either be iterative which is a special variation of the gibbs sampler or be sequential which is a variation of sequential imputation in the latter case one can update the posterior distribution with new observations easily the methods proposed have intuitive statistical implications and can be generalized to accommodate other bayesian like procedures we display some applications of the method in connection with the bayesian bootstrap classification hierarchical models and selection of variables in particular as an application of the method we present a unified treatment of switching regression models driven by a general binary process and we develop a bayesian testing procedure some simulations and a real example are used to illustrate the methods proposed

robust estimation of mean squared error of small area estimators abstract a well known model due to fay and herriot for estimating small area domain means i is considered given i s it is assumed that the survey estimators y i are independent with means i and known variances di i 1 t further the i s are assumed to be independent with means x i and unknown variance a where x i is a vector of benchmark variables related to i and is a vector of regression parameters an empirical best linear unbiased prediction eblup estimator or an empirical linear bayes estimator t i y of i is obtained it is shown that an estimator of mean squared error mse of t t y derived by prasad and rao under normality of i and y i given i is robust with respect to nonnormality of the i s specifically it is shown that the prasad rao estimator of mse is correct to terms of order o t 1 for large t assuming only certain moment conditions on the i s and normally distributed survey errors results of a simulation study on the accur

estimating normal means with a dirichlet process prior abstract in this article the dirichlet process prior is used to provide a nonparametric bayesian estimate of a vector of normal means in the past there have been computational difficulties with this model this article solves the computational difficulties by developing a gibbs sampler algorithm the estimator developed in this article is then compared to parametric empirical bayes estimators peb and nonparametric empirical bayes estimators npeb in a monte carlo study the monte carlo study demonstrates that in some conditions the peb is better than the npeb and in other conditions the npeb is better than the peb the monte carlo study also shows that the estimator developed in this article produces estimates that are about as good as the peb when the peb is better and produces estimates that are as good as the npeb estimator when that method is better

bayesian estimation in the presence of deterministic nuisance parameters part ii estimation methods one of the fundamental issues of estimation theory is the presence of deterministic nuisance parameters while in the bayesian paradigm the model parameters are random introduction of deterministic nuisance parameters into the model exceeds the bayesian framework to the hybrid framework in this type of scenarios the conventional bayesian estimators are not valid as they assume knowledge of the deterministic nuisance parameters this paper is the second of a two part study of bayesian parameter estimation in the presence of deterministic nuisance parameters in part i a new cramer rao cr type bound on the mean square error mse for bayesian estimation in the presence of deterministic nuisance parameters was established based on the concept of risk unbiasedness the proposed bound was named risk unbiased bound rub this paper presents properties of asymptotic uniform mean and risk unbiasedness of some bayesian estimators 1 the minimum mse mmse or maximum a posteriori probability map estimators with maximum likelihood ml estimates substituting the deterministic parameters named ms ml and map ml respectively and 2 joint map and ml estimator named jmap ml furthermore an asymptotic performance analysis of the ms ml and map ml estimators is presented these estimators are shown to asymptotically achieve the rub while the existing cr type bounds can be achieved only in distinct cases simulations verify these results for the problem of blind separation of nonstationary sources it is shown that unlike existing cr type bounds the rub is asymptotically tight

complex gaussian quadrature of oscillatory integrals we construct and analyze gauss type quadrature rules with complex valued nodes and weights to approximate oscillatory integrals with stationary points of high order the method is based on substituting the original interval of integration by a set of contours in the complex plane corresponding to the paths of steepest descent each of these line integrals shows an exponentially decaying behaviour suitable for the application of gaussian rules with non standard weight functions the results differ from those in previous research in the sense that the constructed rules are asymptotically optimal i e among all known methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence relative to the required number of evaluations of the integrand

on measuring the variability of small area estimators under a basic area level model summary in this paper based on a basic area level model we obtain second order accurate approxi mations to the mean squared error of model based small area estimators using the fay herriot 1979 iterative method of estimating the model variance based on weighted residual sum of squares we also obtain mean squared error estimators unbiased to second order based on simulations we compare the finite sample performance of our mean squared error estimators with those based on method of moments maximum likelihood and residual maximum likelihood estimators of the model variance our results suggest that the fay herriot method performs better in terms of relative bias of mean squared error estimators than the other methods across different combinations of number of areas pattern of sampling variances and distribution of small area effects we also derive a noninformative prior on the model parameters for which the posterior variance of a small area mean is second order unbiased for the mean squared error the posterior variance based on such a prior possesses both bayesian and frequentist interpretations

hidden markov models for longitudinal comparisons medical researchers interested in temporal multivariate measurements of complex diseases have recently begun developing health state models which divide the space of patient characteristics into medically distinct clusters the current state of the art in health services research uses k means clustering to form the health states and a first order markov chain to describe transitions between the states this fitting procedure ignores information from temporally adjacent observations and prevents uncertainty from parameter estimation and cluster assignments from being incorporated into the analysis a natural way to address these issues is to combine clustering and longitudinal analyses using a hidden markov model we fit hidden markov models to longitudinal data using bayesian methods that account for all of the uncertainty in the parameters conditional only on the underlying correctness of the model potential lack of time homogeneity in the markov chain is accounted for by embedding transition probabi

badly approximable points on planar curves and a problem of davenport let mathcal c be two times continuously differentiable curve in mathbb r 2 with at least one point at which the curvature is non zero for any i j geqslant 0 with i j 1 let mathbf bad i j denote the set of points x y in mathbb r 2 for which max vert qx vert 1 i vert qy vert 1 j c q for all q in mathbb n here c c x y is a positive constant our main result implies that any finite intersection of such sets with mathcal c has full hausdorff dimension this provides a solution to a problem of davenport dating back to the sixties

hierarchical poisson regression modeling abstract the poisson model and analyses here feature nonexchangeable gamma distributions although exchangeable following a scale transformation for individual parameters with standard deviations proportional to means a relatively uninformative prior distribution for the shrinkage values eliminates the ill behavior of maximum likelihood estimators of the variance components when tested in simulation studies the resulting procedure provides better coverage probabilities and smaller risk than several other published rules and thus works well from bayesian and frequentist perspectives alike the computations provide fast accurate density approximations to individual parameters and to structural regression coefficients the computer program is publicly available through statlib

empirical bayes estimation of the reliability of nuclear power plant emergency diesel generators for a binomial sampling model using pass fail data and a beta prior distribution parametric empirical bayes peb point and credibility interval estimates are provided for the time and diesel average emergency diesel generator edg reliability to load run on demand at 63 u s commercial nuclear power stations for a geometric sampling model and a beta prior distribution similar estimates of the time dependent edg average reliability are also obtained for four plants this is an expository article that introduces seven practical steps involving both model validation and inference in applying modern peb methods it also illustrates some important advantages of using industry wide data to make inferences on individual items

how to be a better bayesian abstract consider an experiment yielding an observable random quantity x whose distribution is indexed by a real parameter o suppose that a statistician is prepared to execute a bayes procedure in estimating o and has quantified available prior information about o in his chosen prior distribution g suppose that before this estimation process is completed the statistician becomes aware of the outcome y of a similar experiment in this article we investigate the questions of whether and when this additional information can be exploited so as to provide a better estimate of o in the current experiment we show that in the traditional empirical bayes framework and in situations involving exponential families conjugate priors and squared error loss the answer is essentially always an explicit bayes empirical bayes beb estimator of o is given that is superior to the original bayes estimator showing that the statistician has in these problems the opportunity to be a better bayesian by combi

estimation of finite population domain means a model assisted empirical best prediction approach in this article we introduce a general methodology for producing a model assisted empirical best predictor ebp of a finite population domain mean using data from a complex survey our method improves on the commonly used design consistent survey estimator by using a suitable mixed model such a model combines information from related sources such as census and administrative data unlike a purely model based ebp the proposed model assisted ebp converges in probability to the customary design consistent estimator as the domain and sample sizes increase the convergence in probability is shown to hold with respect to the sampling design irrespective of the assumed mixed model a property commonly known as design consistency this property ensures robustness of the proposed predictor against possible model failures in addition the convergence in probability is shown to be valid with respect to the assumed mixed model model consistency a new mean squared prediction error mspe estimator is proposed

nonparametric empirical bayes growth curve analysis abstract nonparametric and semiparametric regression have been suggested as alternatives to parametric models for growth curve analysis in this article we demonstrate that empirical bayes estimation can be used to improve linear smoothing estimates when multiple curves are available

badly approximable points on planar curves and winning abstract for any i j 0 with i j 1 let bad i j denote the set of points x y r 2 such that max q x 1 i q y 1 j c q for some positive constant c c x y and all q n we show that bad i j c is winning in the sense of schmidt games for a large class of planar curves c namely everywhere non degenerate planar curves and straight lines satisfying a natural diophantine condition this strengthens recent results solving a problem of davenport from the sixties in short within the context of davenport s problem the winning statement is best possible furthermore we obtain the inhomogeneous generalisations of the winning results for planar curves and lines and also show that the inhomogeneous form of bad i j is winning for two dimensional schmidt games

measuring explained variation in linear mixed effects models we generalize the well known r2 measure for linear regression to linear mixed effects models our work was motivated by a cluster randomized study conducted by the eastern cooperative oncology group to compare two different versions of informed consent document we quantify the variation in the response that is explained by the covariates under the linear mixed model and study three types of measures to estimate such quantities the first type of measures make direct use of the estimated variances the second type of measures use residual sums of squares in analogy to the linear regression the third type of measures are based on the kullback leibler information gain all the measures can be easily obtained from software programs that fit linear mixed models we study the performance of the measures through monte carlo simulations and illustrate the usefulness of the measures on data sets copyright 2003 john wiley sons ltd

a comparison of 20 heterogeneity variance estimators in statistical synthesis of results from studies a simulation study when we synthesize research findings via meta analysis it is common to assume that the true underlying effect differs across studies total variability consists of the within study and between study variances heterogeneity there have been established measures such as i2 to quantify the proportion of the total variation attributed to heterogeneity there is a plethora of estimation methods available for estimating heterogeneity the widely used dersimonian and laird estimation method has been challenged but knowledge of the overall performance of heterogeneity estimators is incomplete we identified 20 heterogeneity estimators in the literature and evaluated their performance in terms of mean absolute estimation error coverage probability and length of the confidence interval for the summary effect via a simulation study although previous simulation studies have suggested the paule mandel estimator it has not been compared with all the available estimators for dichotomous outcomes estimating heterogeneity through markov chain monte carlo is a good choice if an informative prior distribution for heterogeneity is employed eg by published cochrane reviews nonparametric bootstrap and positive dersimonian and laird perform well for all assessment criteria for both dichotomous and continuous outcomes hartung makambi estimator can be the best choice when the heterogeneity values are close to 0 07 for dichotomous outcomes and medium heterogeneity values 0 01 0 05 for continuous outcomes hence there are heterogeneity estimators nonparametric bootstrap dersimonian and laird and positive dersimonian and laird that perform better than the suggested paule mandel maximum likelihood provides the best performance for both types of outcome in the absence of heterogeneity

constrained bayes estimation with applications abstract bayesian techniques are widely used in these days for simultaneous estimation of several parameters in compound decision problems often however the main objective is to produce an ensemble of parameter estimates whose histogram is in some sense close to the histogram of population parameters this is for example the situation in subgroup analysis where the problem is not only to estimate the different components of a parameter vector but also to identify the parameters that are above and the others that are below a certain specified cutoff point we have proposed in this paper bayes estimates in a very general context that meet this need these estimates are obtained by matching the first two moments of the histogram of the estimates and the posterior expectations of the first two moments of the histogram of the parameters and minimizing subject to these conditions the posterior expectation of the euclidean distance between the estimates and the parameters several applications of the m

natural exponential families with quadratic variance functions statistical theory the normal poisson gamma binomial negative binomial and nefghs distributions are the six univariate natural exponential families nef with quadratic variance functions qvf this sequel to morris 1982 treats certain statistical topics that can be handled within this unified nef qvf formulation including unbiased estimation bhattacharyya and cramer rao lower bounds conditional distributions and moments quadratic regression conjugate prior distributions moments of conjugate priors and posterior distributions empirical bayes and g 2 minimax marginal distributions and their moments parametric empirical bayes and characterizations

triple goal estimates in two stage hierarchical models the beauty of the bayesian approach is its ability to structure complicated models inferential goals and analyses to take full advantage of it methods should be linked to an inferential goal via a loss function for example in the two stage compound sampling model the posterior means are optimal under squared error loss however they can perform poorly in estimating the histogram of the parameters or in ranking them triple goal estimates are motivated by the desire to have a set of estimates that produce good ranks a good parameter histogram and good co ordinate specific estimates no set of estimates can simultaneously optimize these three goals and we seek a set that strikes an effective trade off we evaluate and compare three candidate approaches the posterior means the constrained bayes estimates of louis and ghosh and a new approach that optimizes estimation of the histogram and the ranks mathematical and simulation based analyses support the superiority of the new approach and document its excellent performance for the three inferential goals

confidence sets centered at james stein estimators a surprise concerning the unknown variance case abstract we compare the confidence set centered at a james stein point estimator to the usual f confidence set for the p regression parameters of a linear model previous studies usually focused on the known variance case and typically concluded that whatever holds in the known variance case also holds in the unknown variance case when the variance is replaced by its best linear estimator based on s 2 where s 2 is the sum of squarred residuals we are surprised that this is not entirely the scenario we observe here in fact in many situations involving unknown variance the range of the shrinkage factor a for the associated confidence set to have uniformly higher coverage probabilities than its f counterpart can be ten times bigger than 2 p 2 the expected upper bound in the known variance case this is true especially when the degrees of freedom are small our numerical studies also show that to gain substantial improvement one has to use a much larger a especially when the degrees of freedom of the residuals are small application to an economic data set is also included

a random effects regression model for meta analysis many meta analyses use a random effects model to account for heterogeneity among study results beyond the variation associated with fixed effects a random effects regression approach for the synthesis of 2 x 2 tables allows the inclusion of covariates that may explain heterogeneity a simulation study found that the random effects regression method performs well in the context of a meta analysis of the efficacy of a vaccine for the prevention of tuberculosis where certain factors are thought to modify vaccine efficacy a smoothed estimator of the within study variances produced less bias in the estimated regression coefficients the method provided very good power for detecting a non zero intercept term representing overall treatment efficacy but low power for detecting a weak covariate in a meta analysis of 10 studies we illustrate the model by exploring the relationship between vaccine efficacy and one factor thought to modify efficacy the model also applies to the meta analysis of continuous outcomes when covariates are present

parametric empirical bayes inference theory and applications abstract this article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical bayes viewpoint particularly in the case of parametric prior distributions some successful applications of major importance are considered recent results concerning estimates of error and confidence intervals are described and illustrated with data

cantor winning sets and their applications abstract we introduce and develop a class of cantor winning sets that share the same amenable properties as the classical winning sets associated to schmidt s game these include maximal hausdorff dimension invariance under countable intersections with other cantor winning sets and invariance under bi lipschitz homeomorphisms it is then demonstrated that a wide variety of badly approximable sets appearing naturally in the theory of diophantine approximation fit nicely into our broad reaching framework as applications of this phenomenon we answer several previously open questions including some related to the mixed littlewood conjecture and approximation by multiplicative semigroups of integers

parametric empirical bayes tolerance intervals the theory of a parametric empirical bayes peb normal distribution tolerance bound is derived classical tolerance intervals are based on the assumption of a fixed unknown standard deviation and mean such intervals assume no prior knowledge about the possible value of the empirical bayes approach assumes that there is prior knowledge of the value of which can be characterized in terms of a translated beta density with shape parameters and range parameters a b the resulting peb k factors are smaller than those of the classical approach so fewer samples are required to achieve equal precision an example illustrates potential cost savings extensive tables are not provided since the peb k factors depend on a b as well as the sample size confidence and coverage probability since the exact computation is intensive i suggest an approximation based on the noncentral t distribution the degrees of freedom and noncentrality parameters are obtained by matching the first two

comparison of partially measured latent traits across nominal subgroups abstract this article presents a method for estimating the subgroup distributions of a latent trait measured as a normal variate in the population this problem occurs in large scale assessments such as the national assessment of educational progress the national adult literacy survey and other programs where proficiencies are estimated via marginal maximum likelihood through a model based on item response theory heretofore estimates of subgroup means were often estimated using ad hoc assumptions about within group distributions which conflicted with concurrent assumptions about normality of the population distributions the method presented here removes the conflict using consistent distributional assumptions for population and subgroup estimates

bayesian estimation of a multilevel irt model using gibbs sampling in this article a two level regression model is imposed on the ability parameters in an item response theory irt model the advantage of using latent rather than observed scores as dependent variables of a multilevel model is that it offers the possibility of separating the influence of item difficulty and ability level and modeling response variation and measurement error another advantage is that contrary to observed scores latent scores are test independent which offers the possibility of using results from different tests in one analysis where the parameters of the irt model and the multilevel model can be concurrently estimated the two parameter normal ogive model is used for the irt measurement model it will be shown that the parameters of the two parameter normal ogive model and the multilevel model can be estimated in a bayesian framework using gibbs sampling examples using simulated and real data are given

a bayesian approach to ranking and selection of related means with alternatives to analysis of variance methodology abstract a set of unknown normal means treatment effects say 1 2 k is to be investigated two common questions in analysis of variance and ranking and selection are as follows a what is the strength of evidence against the hypothesis h 0 of equality of means b if h 0 is false which mean is the largest or smallest a bayesian approach to the problem is taken leading to calculation of the posterior probability of h 0 and the posterior probabilities that each mean is the largest conditional on h 0 being false a variety of exchangeable nonexchangeable informative and noninformative prior assumptions are considered calculations involve at worst only low dimensional numerical integration in spite of the fact that the dimension k can be arbitrarily large as an example table 1 presents for each baseball team in the national league in 1984 the highest batting average obtained by any player on the team with at least 150 at bats the observed batting averages are treated as sampl

empirical likelihood for small area estimation abstract small areaestimation has become a topic of growingimportance inrecent years due to increasing demand for small area statistics fromboth the public and private sectors the current methodologies inthis area with possibly a handful of exceptions are all parametric be it bayesian or frequentist even the so called nonparamet ric procedures based on empirical best linear unbiased predictors eblup assume linearity of the estimators of small area means resampling methods like jacknife and bootstrap are used mostlyfor mean squared error estimation in this paper we propose analternative bayesian approach based on empirical likelihood whichdoes not require the assumption of either a parametric likelihoodor linearity of estimators the proposed method can also handleboth discrete and continuous data in a uni ed manner empiricallikelihood for both area and unit level models are introduced theimproved performance of our estimator is illustrated through twoexamples involving both discrete and continuous data key words and phrases area level exponentially tilted t prior unit level

badly approximable points on manifolds this paper is motivated by two problems in the theory of diophantine approximation namely davenport s problem regarding badly approximable points on submanifolds of a euclidean space and schmidt s problem regarding the intersections of the sets of weighted badly approximable points the problems have been recently settled in dimension two but remain open in higher dimensions in this paper we develop new techniques that allow us to tackle them in full generality the techniques rest on lattice points counting and a powerful quantitative result of bernik kleinbock and margulis the main theorem of this paper implies that any finite intersection of the sets of weighted badly approximable points on any analytic nondegenerate submanifold of mathbb r n has full dimension one of the consequences of this result is the existence of transcendental real numbers badly approximable by algebraic numbers of any bounded degree

a sample reuse method for accurate parametric empirical bayes confidence intervals abstract parametric empirical bayes methods of point estimation for a vector of unknown parameters date to the landmark paper of james and stein 1961 the usual approach is to use the mean of the estimated posterior distribution of each parameter where the estimation of the prior parameters hyperparameters is accomplished through the marginal distribution of the data while point estimates computed this way usually perform well interval estimates based on the estimated posterior called naive eb intervals are not they fail to account for the variability in the estimation of the hyperparameters generally resulting in sub nominal coverage probability in the eb sense defined in morris 1983a in this paper we extend the work of carlin and gelfand 1989 who proposed a conditional bias correction method for developing eb intervals which corrects the deficiencies in the naive intervals we show how bias correction can be implemented in general via a type iii parametric bootstrap procedure a sample reuse method first employed by laird and louis 1987

estimating heterogeneity variance in meta analysis summary several new estimators of the between study variability in a heterogeneous random effects meta analysis model are derived one is the unbiased statistic which is locally optimal for small values of the parameter others are bayes procedures within a class of quadratic statistics for a diffuse prior with different choices of the prior mean these estimators are compared with the dersimonian laird procedure and the hedges statistic in particular via the quadratic risk of the treatment effect estimator a monte carlo study supports the usage of confidence intervals derived from the new estimators

estimation of a stationary markov chain this article addresses the issue of empirical bayes estimation procedures for the transition probability matrix of a single stationary markov chain the distribution of the frequency count matrix is obtained by relying heavily on the whittle distributions and on the natural conjugate priors thence empirical bayes estimates are obtained and the limiting distribution is derived

toward a reconciliation of the bayesian and frequentist approaches to point estimation abstract the bayesian and frequentist approaches to point estimation are reviewed the status of the debate regarding the use of one approach over the other is discussed and its inconclusive character is noted a criterion for comparing bayesian and frequentist estimators within a given experimental framework is proposed the competition between a bayesian and a frequentist is viewed as a contest with the following components a random observable a true prior distribution unknown to both statisticians an operational prior used by the bayesian a fixed frequentist rule used by the frequentist and a fixed loss criterion this competition is studied in the context of exponential families conjugate priors and squared error loss the class of operational priors that yield bayes estimators superior to the best frequentist estimator is characterized the implications of the existence of a threshold separating the space of operational priors into good and bad priors are explored and their relevance in ar

multiplicatively badly approximable numbers and generalised cantor sets abstract let p be a prime number the p adic case of the mixed littlewood conjecture states that l i m i n f q q q p q 0 for all r we show that with the additional factor of log q log log q the statement is false indeed our main result implies that the set of for which l i m i n f q q log q log log q q p q 0 is of full dimension the result is obtained as an application of a general framework for cantor sets developed in this paper

a note on the empirical bayes heterogeneity variance estimator in meta analysis this paper focuses on the empirical bayes eb or mandel paule estimator of the heterogeneity variance in meta analysis which was discussed by morris and proposed in earlier publications by mandel and paule in an inter laboratory context the relationship of the eb estimator to other heterogeneity variance estimators typically used in meta analysis is explored and approximate variance estimators for the eb estimate of the heterogeneity variance are proposed based on the m estimation method statistical inference for the overall treatment effect using the eb estimator and the proposed standard errors is discussed using two example data sets from meta analysis applications

frequentist accuracy of bayesian estimates in the absence of relevant prior experience popular bayesian estimation techniques usually begin with some form of uninformative prior distribution intended to have minimal inferential influence bayes rule will still produce nice looking estimates and credible intervals but these lack the logical force attached to experience based priors and require further justification this paper concerns the frequentist assessment of bayes estimates a simple formula is shown to give the frequentist standard deviation of a bayesian point estimate the same simulations required for the point estimate also produce the standard deviation exponential family models make the calculations particularly simple and bring in a connection to the parametric bootstrap

on a problem in simultaneous diophantine approximation schmidt s conjecture for any i j 0 with i j 1 let bad i j denote the set of points x y 2 r 2 for which maxfkqxk 1 i kqyk 1 j g c q for all q 2 n here c c x y is a positive constant our main result implies that any nite intersection of such sets has full dimension this settles a conjecture of wolfgang m schmidt in the theory of simultaneous diophantine approximation

on some n 1 symmetric linear functionals abstract many authors have dealt with problems related to the symmetrization of sequences of orthogonal polynomials on a real line or on a unit circle particular aspects are treated for quadratic or cubic decompositions in this paper we present a technique that unifies the treatment of these topics for an arbitrary positive order of decomposition n by considering a more general definition of orthogonality this technique is based on the decomposition with respect to the cyclic group of order n the main theorem is applied to the orthogonality on n 1 symmetric curves some particular cases are singled out these results may be useful in studying hermite pade approximations vector continued fractions and dynamical systems

general maximum likelihood empirical bayes estimation of normal means we propose a general maximum likelihood empirical bayes gmleb method for the estimation of a mean vector based on observations with i i d normal errors we prove that under mild moment conditions on the unknown means the average mean squared error mse of the gmleb is within an infinitesimal fraction of the minimum average mse among all separable estimators which use a single deterministic estimating function on individual observations provided that the risk is of greater order than log n 5 n we also prove that the gmleb is uniformly approximately minimax in regular and weak l p balls when the order of the length normalized norm of the unknown means is between log n k1 n 1 p 2 and n log n k2 simulation experiments demonstrate that the gmleb outperforms the james stein and several state of the art threshold estimators in a wide range of settings without much down side

intrinsic diophantine approximation on manifolds general theory we investigate the question of how well points on a nondegenerate k dimensional submanifold m subseteq mathbb r d can be approximated by rationals also lying on m establishing an upper bound on the intrinsic dirichlet exponent for m we show that relative to this exponent the set of badly intrinsically approximable points is of full dimension and the set of very well intrinsically approximable points is of zero measure our bound on the intrinsic dirichlet exponent is phrased in terms of an explicit function of k and d which does not seem to have appeared in the literature previously it is shown to be optimal for several particular cases the requirement that the rationals lie on m distinguishes this question from the more common context of ambient diophantine approximation on manifolds and necessitates the development of new techniques our main tool is an analogue of the simplex lemma for rationals lying on m which provides new insights on the local distribution of rational points on nondegenerate manifolds

combining forecasts operational adjustments to theoretically optimal rules clemen and winkler 1985 have described the theoretical effectiveness of winkler s 1981 formula for optimally combining forecasts the optimality of winkler s formula is however contingent on actually knowing the forecasters statistical properties i e the variances and covariances of their forecasts in realistic applications of course these properties have to be estimated usually from a set of prior forecasts in this case we show how the operationally optimal combining strategy differs from winkler s theoretically optimal formula specifically we provide figures indicating the operationally optimal strategy for combining two forecasts we then propose a heuristic to choose the best set of parameter estimates in combining any number of forecasters and demonstrate its effectiveness via simulation

comparisons of alternative predictors under the balanced one way random model abstract prediction of an arbitrary linear combination of the random effects of a balanced one way random model is investigated alternative two stage predictors are compared on the basis of their conditional on the random effects and unconditional bias and mean squared errors when the true value of the ratio of expected mean squares is known there exists a best linear unbiased predictor blup when the true value is unknown a two stage predictor obtained from the blup by replacing the true value with an estimated value can be used when the ratio of expected mean squares is estimated by maximum likelihood bayesian methods or various related methods a two stage predictor is obtained whose properties compare favorably with for example those of the least squares predictor and the positive part james stein predictor

shrinkage estimation in the two way multivariate normal model a two way multivariate normal model is proposed and attention is focused on estimation of the mean values when the common variance of the observations is unknown a class of empirical bayes estimators is proposed and mean squared errors are given a lower bound on the mean squared error is found and related to risk asymptotics a jamesstein type estimator is derived and compared with its competitor a modal estimator that is obtained from a hierarchical prior for the unknown parameters

monotone empirical bayes estimators based on more informative samples abstract an empirical bayes estimator is constructed for the case of nonidentical components coming from an exponential family of distributions that is closed under convolution examples are the normal poisson and binomial distributions the construction is based on robbins approach to empirical bayes estimation and generalizes results of o bryan and stijnen and van houwelingen the method is elaborated for the normal and poisson case and examples are given that show that this approach gives results comparable to those from the nonparametric maximum likelihood approach of laird

improved estimation of subject level functional connectivity using full and partial correlation with empirical bayes shrinkage abstract reliability of subject level resting state functional connectivity fc is determined in part by the statistical techniques employed in its estimation methods that pool information across subjects to inform estimation of subject level effects e g bayesian approaches have been shown to enhance reliability of subject level fc however fully bayesian approaches are computationally demanding while empirical bayesian approaches typically rely on using repeated measures to estimate the variance components in the model here we avoid the need for repeated measures by proposing a novel measurement error model for fc describing the different sources of variance and error which we use to perform empirical bayes shrinkage of subject level fc towards the group average in addition since the traditional intra class correlation coefficient icc is inappropriate for biased estimates we propose a new reliability measure denoted the mean squared error intra class correlation coefficient iccmse to properly assess the reliability of the resulting biased estimates we apply the proposed techniques to test retest resting state fmri data on 461 subjects from the human connectome project to estimate connectivity between 100 regions identified through independent components analysis ica we consider both correlation and partial correlation as the measure of fc and assess the benefit of shrinkage for each measure as well as the effects of scan duration we find that shrinkage estimates of subject level fc exhibit substantially greater reliability than traditional estimates across various scan durations even for the most reliable connections and regardless of connectivity measure additionally we find partial correlation reliability to be highly sensitive to the choice of penalty term and to be generally worse than that of full correlations except for certain connections and a narrow range of penalty values this suggests that the penalty needs to be chosen carefully when using partial correlations

inference and prediction for a general order statistic model with unknown population size abstract suppose that the first n order statistics from a random sample of n positive random variables are observed where n is unknown this the general order statistic model has been applied to the study of market penetration capture recapture burn in in repairable systems software reliability growth the estimation of the number of individuals exposed to radiation and the estimation of the number of unseen species inference is to be made about the unknown parameters especially n and future observations are to be predicted a bayes empirical bayes approach to inference is presented this permits the comparison of competing perhaps nonnested models for the distribution of the random variables in a natural way it also provides easily implemented inference and prediction procedures that avoid the difficulties of non bayesian methods one such difficulty is that the maximum likelihood estimator of n may be infinite results are given for the case in which vague prior information about the model

robust bayes and empirical bayes analysis with _ epsilon contaminated priors on considere deux directions dans l analyse bayesienne robuste la premiere consiste a determiner l etendue des probabilites a posteriori d un ensemble la seconde consiste a selectionner une bonne distribution a priori

multistep estimators of the between study variance the relationship with the paule mandel estimator a wide variety of estimators of the between study variance are available in random effects meta analysis many but not all of these estimators are based on the method of moments the dersimonian laird estimator is widely used in applications but the paule mandel estimator is an alternative that is now recommended recently dersimonian and kacker have developed two step moment based estimators of the between study variance we extend these two step estimators so that multiple more than two steps are used we establish the surprising result that the multistep estimator tends towards the paule mandel estimator as the number of steps becomes large hence the iterative scheme underlying our new multistep estimator provides a hitherto unknown relationship between two step estimators and paule mandel estimator our analysis suggests that two step estimators are not necessarily distinct estimators in their own right instead they are quantities that are closely related to the usual iterative scheme that is used to calculate the paule mandel estimate the relationship that we establish between the multistep and paule mandel estimator is another justification for the use of the latter estimator two step and multistep estimators are perhaps best conceptualized as approximate paule mandel estimators

a james stein type estimator for combining unbiased and possibly biased estimators abstract we present a method for combining unbiased sample data with possibly biased auxiliary information the estimator we derive is similar in spirit to the james stein estimator we prove that the estimator dominates the sample mean under quadratic loss when the auxiliary information is unbiased our estimator has risk slightly greater than the usual combined estimator as the bias increases however the risk of the usual estimator is unbounded while the risk of our estimator is bounded by the risk of the sample mean we show how our estimator can be considered an approximation to the best linear combination of the sample data and the auxiliary information allude to how it can be derived as an empirical bayes estimator and suggest a method for constructing confidence sets finally the performance of our estimator is compared to that of the sample mean and the usual combined estimator using real forestry data

using empirical bayes methods in biopharmaceutical research a compound sampling model where a unit specific parameter is sampled from a prior distribution and then observed are generated by a sampling distribution depending on the parameter underlies a wide variety of biopharmaceutical data for example in a multi centre clinical trial the true treatment effect varies from centre to centre observed treatment effects deviate from these true effects through sampling variation knowledge of the prior distribution allows use of bayesian analysis to compute the posterior distribution of clinic specific treatment effects frequently summarized by the posterior mean and variance more commonly with the prior not completely specified observed data can be used to estimate the prior and use it to produce the posterior distribution an empirical bayes or variance component analysis in the empirical bayes model the estimated prior mean gives the typical treatment effect and the estimated prior standard deviation indicates the heterogeneity of treatment effects in both the bayes and empirical bayes approaches estimated clinic effects are shrunken towards a common value from estimates based on single clinics this shrinkage produces more efficient estimates in addition the compound model helps structure approaches to ranking and selection provides adjustments for multiplicity allows estimation of the histogram of clinic specific effects and structures incorporation of external information this paper outlines the empirical bayes approach coverage will include development and comparison of approaches based on parametric priors for example a gaussian prior with unknown mean and variance and non parametric priors discussion of the importance of accounting for uncertainty in the estimated prior comparison of the output and interpretation of fixed and random effects approaches to estimating population values estimating histograms and identification of key considerations in the use and interpretation of empirical bayes methods

innovations in bayes and empirical bayes methods estimating parameters populations and ranks by formalizing the relation among components and borrowing information among them bayes and empirical bayes methods can produce more valid efficient and informative statistical evaluations than those based on traditional methods in addition bayesian structuring of complicated models and goals guides development of appropriate statistical approaches and generates summaries which properly account for sampling and modelling uncertainty computing innovations enable implementation of complex and relevant models thereby substantially increasing the role of bayes empirical bayes methods in important statistical assessments policy relevant statistical assessments involve synthesis of information from a set of related components such as medical clinics geographic regions or research studies typical assessments include inference for individual parameters synthesis over the collection of components for example the parameter histogram and comparisons among parameters for example ranks the relative importance of these goals depends on the context bayesian structuring provides a guide to valid inference for example while posterior means are the obvious and optimal estimates for individual components under squared error loss their empirical distribution function edf is underdispersed and never valid for estimating the edf of the true underlying parameters effective histogram estimates result from optimizing a loss function based in a distance between the histogram and its estimate similarly ranking observed data usually produces poor estimates and ranking posterior means can be inappropriate effective estimates should be based on a loss function that caters directly to ranks using examples of borrowing information shrinkage and the variance bias trade off we motivate bayes and empirical bayes analysis then we outline the formal approach and discuss triple goal estimates with values that when ranked produce optimal ranks for which the edf is an optimal estimate of the parameter edf and such that the values themselves are effective estimates of co ordinate specific parameters we use basic models and data analysis examples to highlight the conceptual and structural issues

priors about observables in vector autoregressions we formulate a prior about observables in a vector autoregression var and then solve the deconvolution problem for the implied prior about var parameters formulating a prior about observables is more intuitive than formulating a prior about var parameters directly because var parameters are hard to interpret our numerical algorithm for approximating the implied prior about parameters works well even in high dimensional problems and can be applied also for models other than vars in the empirical application we formulate a prior about growth rates of the observables in a var model of the united states economy we nd that this prior makes a big dierence for the estimated persistence of output responses to monetary policy shocks compared with the results of standard priors for vars

a class of orthogonal polynomials on the radial rays in the complex plane abstract we introduce a class of polynomials orthogonal on some radial rays in the complex plane and investigate their existence and uniqueness a recurrence relation for these polynomials a representation and the connection with standard polynomials orthogonal on 0 1 are derived it is shown that their zeros are simple and distributed symmetrically on the radial rays with the possible exception of a multiple zero at the origin an analogue of the jacobi polynomials and the corresponding problem with the generalized laguerre polynomials are also treated

a model for incorporating historical controls into a meta analysis a method is presented for estimating the treatment effect in a meta analysis when some of the studies are comparative studies and the rest are noncomparative historical control studies a random effects model is used in which the baseline effect in each study is random but the treatment effect is constant with this model the appropriate contribution of the historical studies can be determined extensions of the model are developed to accommodate preliminary tests for bias and for the possibility that the treatment effect is heterogeneous

empirical bayes confidence sets for the mean of a multivariate normal distribution abstract through the use of an empirical bayes argument a confidence set for the mean of a multivariate normal distribution is derived the set is a recentered sphere is easy to compute and has uniformly smaller volume than the usual confidence set an exact formula for the coverage probability is derived and numerical evidence is presented which shows that the empirical bayes set uniformly dominates the usual set in coverage probability

sharp simultaneous confidence intervals for the means of selected populations with application to microarray data analysis simultaneous inference for a large number n of parameters is a challenge in some situations such as microarray experiments researchers are only interested in making inference for the k parameters corresponding to the k most extreme estimates hence it seems important to construct simultaneous confidence intervals for these k parameters the naive simultaneous confidence intervals for the k means applied directly without taking into account the selection have low coverage probabilities we take an empirical bayes approach or an approach based on the random effect model to construct simultaneous confidence intervals with good coverage probabilities for n 10 000 and k 100 typical for microarray data our confidence intervals could be 77 shorter than the naive k dimensional simultaneous intervals

a semi bayes approach to the analysis of correlated multiple associations with an application to an occupational cancer mortality study thomas et al presented the application of empirical bayes methods to the problem of multiple inference in epidemiologic studies one limitation of their approach which they noted was the need to assume exchangeable log relative risk parameters and independent relative risk estimates numerical integration was also required here i generalize their approach to allow for non exchangeable parameters and non independent estimates the resulting method is bayesian in so far as some feature of the prior distribution are specified from prior information but is empirical bayes in so far as some explicit parameters in the prior distribution are estimated from the data estimation is based on approximations to the posterior distribution this allows one to implement the approach with standard software packages for matrix algebra the method is illustrated in an occupational mortality study of 84 exposure cancer associations

empirical bayes methods for combining likelihoods abstract suppose that several independent experiments are observed each one yielding a likelihood l k k for a real valued parameter of interest k for example k might be the log odds ratio for a 2 2 table relating to the kth population in a series of medical experiments this article concerns the following empirical bayes question how can we combine all of the likelihoods l k to get an interval estimate for any one of the k s say 1 the results are presented in the form of a realistic computational scheme that allows model building and model checking in the spirit of a regression analysis no special mathematical forms are required for the priors or the likelihoods this scheme is designed to take advantage of recent methods that produce approximate numerical likelihoods l k k even in very complicated situations with all nuisance parameters eliminated the empirical bayes likelihood theory is extended to situations where the k s have a regression structure as well as an empiri

orthogonal matrix polynomials and higher order recurrence relations abstract it is well known that orthogonal polynomials on the real line satisfy a three term recurrence relation and conversely every system of polynomials satisfying a three term recurrence relation is orthogonal with respect to some positive borel measure on the real line we extend this result and show that every system of polynomials satisfying some 2 n 1 term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are n n matrices we apply this result to polynomials orthogonal with respect to a discrete sobolev inner product and other inner products in the linear space of polynomials as an application we give a short proof of krein s characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points

empirical bayes analysis for systems of mixed models with linked autocorrelated random effects abstract empirical bayes concepts are implemented in a simultaneous analysis of a system of mixed linear models having linked and serially correlated random effects emphasis is placed on the estimation of the random effects and exploration of the relationships between them application is made to the investigation of several series of laboratory assay data that were observed during overlapping time intervals and were therefore subjected to common systematic errors or daily effects the motivation for this work was the need to investigate methods of adjustment for such daily effects and to estimate the degree to which concurrently run series are impacted in common attention is given to the construction of confidence intervals for daily effects tractable methods are proposed that yield approximately correct coverage for large samples although derived within a bayes empirical bayes framework these intervals are somewhat similar to intervals constructed by the method of kackar and harville implement

robust empirical bayes analyses of event rates a collection of i similar items generates point event histories for example machines experience failures or operators make mistakes suppose the intervals between events are modeled as iid exponential i or the counts as poisson i t i for the ith item furthermore so as to represent between item variability each individual rate parameter i is presumed drawn from a fixed super population with density g being a vector parameter a parametric empirical bayes peb setup for g specified alternatively as log student t n or gamma we exhibit the results of numerical procedures for estimating superpopulation parameters ll and for describing pooled estimates of the individual rates i obtained via bayes s formula three data sets are analyzed and convenient explicit approximate formulas are furnished for i estimates in the student t case the individual estimates are seen to have a robust quality

admissible estimation dirichlet principles and recurrence of birth death chains on ℤ p we establish a connection between admissible simultaneous estimation and recurrence of reversible markov chains on specifically to each generalized bayes estimator of the mean of a vector of p independent poisson variables for a weighted quadratic loss we associate a variational problem and a reversible birth and death chain on the variational problem is closely related to the dirichlet principle for reversible chains studied recently by griffeath liggett and lyons under side conditions admissibility of the estimator is equivalent to zero infimal energy in the variational problem and to recurrence of the markov chain this yields analytic and probabilistic criteria for inadmissibility which are applied to establish a broad class of results and previous conjectures