on a decomposition lemma for positive semi definite block matrices abstract this short note in part of expository nature points out several new or recent consequences of a quite nice decomposition for positive semi definite matrices

empirical bayes false coverage rate controlling confidence intervals summary benjamini and yekutieli suggested that it is important to account for multiplicity correction for confidence intervals when only some of the selected intervals are reported they introduced the concept of the false coverage rate fcr for confidence intervals which is parallel to the concept of the false discovery rate in the multiple hypothesis testing problem and they developed confidence intervals for selected parameters which control the fcr their approach requires the fcr to be controlled in the frequentist s sense i e controlled for all the possible unknown parameters in modern applications the number of parameters could be large as large as tens of thousands or even more as in microarray experiments we propose a less conservative criterion the bayes fcr and study confidence intervals controlling it for a class of distributions the bayes fcr refers to the average fcr with respect to a distribution of parameters under such a criterion we propose some confidence intervals which by some analytic and numerical calculations are demonstrated to have the bayes fcr controlled at level q for a class of prior distributions including mixtures of normal distributions and zero where the mixing probability is unknown the confidence intervals are shrinkage type procedures which are more efficient for the is that have a sparsity structure which is a common feature of microarray data more importantly the centre of the proposed shrinkage intervals reduces much of the bias due to selection consequently the proposed empirical bayes intervals are always shorter in average length than the intervals of benjamini and yekutieli and can be only 50 or 60 as long in some cases we apply these procedures to the data of choe and colleagues and obtain similar results

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

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

increasing the confidence in student s t interval the usual confidence interval based on student s t distribution has conditional confidence that is larger than the nominal confidence level although this fact is known along with the fact that increased conditional confidence can be used to improve a confidence assertion the confidence assertion of student s t interval has never been critically examined we do so here and construct a confidence estimator that allows uniformly higher confidence in the interval and is closer than 1 a to the indicator of coverage 1 introduction and summary the usual confidence interval for a normal mean when the population variance is unknown is based on student s

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

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

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

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

simultaneous confidence regions for multivariate bioequivalence demonstrating bioequivalence of several pharmacokinetic pk parameters such as auc and cmax that are calculated from the same biological sample measurements is in fact a multivariate problem even though this is neglected by most practitioners and regulatory bodies who typically settle for separate univariate analyses we believe however that a truly multivariate evaluation of all pk measures simultaneously is clearly more adequate in this paper we review methods to construct joint confidence regions around multivariate normal means and investigate their usefulness in simultaneous bioequivalence problems via simulation some of them work well for idealised scenarios but break down when faced with real data challenges such as unknown variance and correlation among the pk parameters we study the shapes of the confidence regions resulting from different methods discuss how marginal simultaneous confidence intervals for the individual pk measures can be derived and illustrate the application to data from a trial on ticlopidine hydrochloride an r package is available

inference for multivariate normal hierarchical models this paper provides a new method and algorithm for making inferences about the parameters of a two level multivariate normal hierarchical model one has observed j p dimensional vector outcomes distributed at level 1 as multivariate normal with unknown mean vectors and with known covariance matrices at level 2 the unknown mean vectors also have normal distributions with common unknown covariance matrix a and with means depending on known covariates and on unknown regression coefficients the algorithm samples independently from the marginal posterior distribution of a by using rejection procedures functions such as posterior means and covariances of the level 1 mean vectors and of the level 2 regression coefficient are estimated by averaging over posterior values calculated conditionally on each value of a drawn this estimation accounts for the uncertainty in a unlike standard restricted maximum likelihood empirical bayes procedures it is based on independent draws from the exact posterior distributions unlike gibbs sampling the procedure is demonstrated for profiling hospitals based on patients responses concerning p 2 types of problems non surgical and surgical the frequency operating characteristics of the rule corresponding to a particular vague multivariate prior distribution are shown via simulation to achieve their nominal values in that setting

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

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

improved confidence sets for the coefficients of a linear model with spherically symmetric errors on obtient des resultats de domination semblables a ceux de hwang et casella 1984 pour differentes distributions a symetrie spherique

small confidence sets for the mean of a spherically symmetric distribution summary suppose that x has a k variate spherically symmetric distribution with mean vector 0 and identity covariance matrix wepresent two spherical confidence sets for 0 both centred at a positive part stein estimator t x in the first we obtain the radius by approximating the upper a point of the sampling disiribution of iits x 9112 by the first two non zero terms of its taylor series about the origin we can analyse some of the properties of this confidence set and see that it performs well in terms of coverage probability volume and conditional behaviour in the second method we find the radius by using a parametric bootstrap procedure here even greater improvement in terms of volume over the usual confidence set is possible at the expense of having a less explicit radius function a real data example is provided and extensions to the unknown covariance matrix and elliptically symmetric cases are discussed

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

inequalities and equalities for the cartesian decomposition of complex matrices let a and b be hermitian matrices and let c a ib inequalities and equalities for the eigenvalues and singular values of the matrices a b and c are discussed known results on inequalities are surveyed new results on equality cases are proved and open problems are mentioned

small divisors with spatial structure in infinite dimensional hamiltonian systems a general perturbation theory of the kolmogorov arnold moser type is described concerning the existence of infinite dimensional invariant tori in nearly integrable hamiltonian systems the key idea is to consider hamiltonians with aspatial structure and to express all quantitative aspects of the theory in terms of rather general weight functions on such structures this approach combines great flexibility with an effective control of the vrious interactions in infinite dimensional systems

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

bounds on the trajectories of a system of weakly coupled rotators in this paper we study a classical mechanical system of weakly coupled rotators on a one dimensional lattice such systems are of interest in statistical mechanics we prove that for any site in the system there is a large set of initial conditions for which there exists a canonical change of variables such that the trajectory of that site in the transformed system is essentially indistinguishable from that of an integrable system for a long but finite time alternatively the trajectory of this site lies very close to a torus in the phase space of the system for times very long in comparison with the typical period of the unperturbed rotators all the estimates in this theory areindependent of the number of degrees of freedom in the system we propose this mechanism as an explanation of certain numerical experiments

on the elimination of non resonance harmonics given a weakly coupled hamiltonian system with short range one dimensional interactions andany initial conditions a canonical change of variables is constructed which yields a new hamiltonian consisting of three parts an integrable term a resonant term whose effects are localized in those regions of the system which give small denominators in the kolmogorov arnol d moser iteration scheme and a non resonant interaction term which is very small in particular much much smaller than our original interactions the conditions which allow such a transformation to be constructed are independent of the number of degrees of freedom in the system as are the estimates on the size of the various terms thus if the resonances are sparsely distributed through the system most of the sites in the transformed hamiltonian behave essentially like an integrable system at least for as long a time as the trajectory of the system lies within the region where the canonical transformation is defined in subsequent work it is shown that this time is long and once again independent of the number of degrees of freedom in the system

on the problem of energy equipartition for large systems of the fermi pasta ulam type analytical and numerical estimates abstract we report on some analytical and numerical results on the exchanges of energy in systems of the fermi pasta ulam type in the light of nekhoroshev s theorem with particular attention to the dependence of the estimates on the number n of degrees of freedom for the ordinary fpu problem we look for a control of the single normal mode energies and we find both the analytical and numerical estimates to agree in predicting that the energy exchanges of the single modes cannot be controlled in the thermodynamic limit we consider then a modified fpu model with alternating light and heavy particles which appears as composed of two subsystems of low acoustic frequency and of high optical frequency respectively we try to control the exchange of the total energy of the high frequency modes up to times increasing exponentially with the frequency in this case the numerical estimates are stronger than the available analytical ones and give indications for nonequipartition with constants essentially independent of the number n of degrees of freedom

minimum volume confidence regions for a multivariate normal mean vector summary since stein s original proposal in 1962 a series of papers have constructed confidence regions of smaller volume than the standard spheres for the mean vector of a multivariate normal distribution a general approach to this problem is developed here and used to calculate a lower bound on the attainable volume bayes and fiducial methods are involved in the calculation scheffe type problems are used to show that low volume by itself does not guarantee favourable inferential properties

factorizing complex symmetric matrices with positive definite real and imaginary parts complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for lu factorization this result adds to the classes of matrix for which it is known to be safe not to pivot in lu factorization block mathrm ldl t factorization with the pivoting strategy of bunch and kaufman is also considered and it is shown that for such matrices only 1 times 1 pivots are used and the same growth factor bound of 2 holds but that interchanges that destroy band structure may be made the latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in linpack and lapack

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

admissibility of the usual confidence set for the mean of a univariate or bivariate normal population the unknown variance case summary in the gaussian linear regression model with unknown mean and variance we show that the standard confidence set for one or two regression coefficients is admissible in the sense of joshi this solves a long standing open problem in mathematical statistics and this has important implications on the performance of modern inference procedures post model selection or post shrinkage particularly in situations where the number of parameters is larger than the sample size as a technical contribution of independent interest we introduce a new class of conjugate priors for the gaussian location scale model

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

the singular values of a b and a ib we obtain several inequalities relating the singular values of a b and those of a ib when a and b are hermitian and when one or both of them are positive semidefinite

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

optimal confidence sets bioequivalence and the limacon of pascal abstract we begin with a decision theoretic investigation into confidence sets that minimize expected volume at a given parameter value such sets are constructed by inverting a family of uniformly most powerful tests and hence they also enjoy the optimality property of being uniformly most accurate in addition these sets possess bayesian optimal volume properties and represent the first case to our knowledge of a frequentist 1 confidence set that possesses a bayesian optimality property the hypothesis testing problem that generates these sets is similar to that encountered in bioequivalence testing our sets are optimal for testing bioequivalence in certain settings in the case of the normal distribution the optimal set is a curve known as the limacon of pascal we illustrate the use of these curves with a biopharmaceutical example

construction of analytic kam surfaces and effective stability bounds a class of analytic possibly time dependent hamiltonian systems withd degrees of freedom and the corresponding class of area preserving twist diffeomorphisms of the plane are considered implementing a recent scheme due to moser salamon and zehnder we provide a method that allows us to construct explicitly kam surfaces and hence to give lower bounds on their breakdown thresholds we then apply this method to the hamiltonianh y2 2 e cosx cos x t and to the map y x y e sinx x y e sinx obtaining with the aid of computer assisted estimations explicit approximations within an error of 10 5 of the golden mean kam surfaces for complex values of e with e less or equal than respectively 0 015 and 0 65 the experimental numerical values at which such surfaces are expected to disappear are about respectively 0 027 and 0 97 a possible connection between break down thresholds and singularities in the complex e plane is pointed out

fischer type determinantal inequalities for accretive dissipative matrices abstract let a a 11 a 12 a 21 a 22 be an n n accretive dissipative matrix k and l be the orders of a 11 and a 22 respectively and let m min k l then det a a det a 11 det a 22 where a 2 3 m 2 if m n 3 2 n 2 if n 3 m n 2 this improves a result of ikramov

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

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

good exact confidence sets for a multivariate normal mean a class of confidence sets with constant coverage probability for the mean of a p variate normal distribution is proposed through a pseudo empirical bayes construction when the dimension is greater than 2 by combining analytical results with some exact numerical calculations the proposed sets are proved to have a uniformly smaller volume than the usual confidence region sufficient conditions for the connectedness of the proposed confidence sets are also derived in addition our confidence sets could be used to construct tests for point null hypotheses the resultant tests have convex acceptance regions and hence are admissible by birnbaum tabular results of the comparison between the proposed region and other confidence sets are also given

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

norm inequalities for cartesian decompositions abstract let the cartesian decomposition of a complex n n matrix t be t a ib with a b hermitian let j and j be the eigenvalues of a and b respectively ordered so that 1 n and 1 n we prove that diag 1 i 1 n i n 2 t for every unitarily invariant norm this settles affirmatively a conjecture of ando and bhatia t ando r bhatia eigenvalue inequalities associated with the cartesian decomposition linear and multilinear algebra 22 1987 133

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

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

norm inequalities for accretive dissipative operator matrices abstract an operator on a hilbert space is accretive dissipative if its real and imaginary parts are both positive in this paper we consider norm inequalities for an accretive dissipative operator matrix the inequalities obtained extend some known results

lieb functions and sectorial matrices abstract in this article we present a special treatment of sectorial matrices under lieb functions this treatment results from a new characterization of sectorial matrices using blocks which then implies a new set of inequalities governing such matrices where singular values determinants numerical radius and unitarily invariant norms inequalities are retrieved in a general form that implies many well known results in the literature

norm inequalities for sector block matrices abstract we study unitarily invariant norm inequalities for sector block matrices and obtain new inequalities which extend some recent results of jabbarzadeh and kaleibary 2020 and generalize a norm inequality due to lin and zhou 2013 at the same time we refine an inequality raised by zhang 2015

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

the kam theory of systems with short range interactions i the existence of quasiperiodic trajectories for hamiltonian systems consisting of long chains of nearly identical subsystems with interactions which decay rapidly with increasing distance between the interacting components is studied such models are of interest in statistical mechanics it is shown that nonergodic motions persist for much larger perturbations than prior work indicated if the number of degrees of freedom of the system isn the allowed perturbation decreases only as an inverse power ofn as the number of degrees of freedom increases rather than the inverse power ofn which previous estimates yielded

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

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

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

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

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

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 intervals shrinking both means and variances summary we construct empirical bayes intervals for a large number p of means the existing intervals in the literature assume that variances are either equal or unequal but known when the variances are unequal and unknown the suggestion is typically to replace them by unbiased estimators however when p is large there would be advantage in borrowing strength from each other we derive double shrinkage intervals for means on the basis of our empirical bayes estimators that shrink both the means and the variances analytical and simulation studies and application to a real data set show that compared with the t intervals our intervals have higher coverage probabilities while yielding shorter lengths on average the double shrinkage intervals are on average shorter than the intervals from shrinking the means alone and are always no longer than the intervals from shrinking the variances alone also the intervals are explicitly defined and can be computed immediately

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 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

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

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

norm inequalities involving accretive dissipative 2 2 block matrices abstract let t 11 t 12 t 21 and t 22 be n n complex matrices and let t t 11 t 12 t 21 t 22 be accretive dissipative it is shown that if f is an increasing convex function on 0 such that f 0 0 then f t 12 2 f t 21 2 f t 2 for every unitarily invariant norm moreover if f is an increasing concave function on 0 such that f 0 0 then f t 12 2 f t 21 2 4 f t 2 4 for every unitarily invariant norm among other inequalities for the schatten p norms it is shown that t 12 p p t 21 p p 2 p 1 t 11 p p 2 t 22 p p 2 for p 2