modified signed log likelihood ratio summary the signed log likelihood ratio r for a one dimensional interest parameter can be modified as r r r log u r so that r is asymptotically standard normally distributed with error of order o n 3 2 u being interpretable as a test statistic a new more direct derivation of this result is given which leads to a more explicit expression for u and provides a better handle on the error term the relation of the tail area approximation determined by r to a generalization of the lugannani rice approximation is discussed and the applicability of r to the analysis of residual variation is indicated

fast and accurate binary response mixed model analysis via expectation propagation expectation propagation is a general prescription for approximation of integrals in statistical inference problems its literature is mainly concerned with bayesian inference scenarios however ex

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

approaches for empirical bayes confidence intervals abstract parametric empirical bayes eb methods of point estimation date to the landmark paper by james and stein 1961 interval estimation through parametric empirical bayes techniques has a somewhat shorter history which was summarized by laird and louis 1987 in the exchangeable case one obtains a naive eb confidence interval by simply taking appropriate percentiles of the estimated posterior distribution of the parameter where the estimation of the prior parameters hyperparameters is accomplished through the marginal distribution of the data unfortunately these naive intervals tend to be too short since they fail to account for the variability in the estimation of the hyperparameters that is they do not attain the desired coverage probability in the eb sense defined by morris 1983a b they also provide no statement of conditional calibration rubin 1984 in this article we propose a conditional bias correction method for developing em intervals that corrects these deficiencies

similar tests and the standardized log likelihood ratio statistic summary when testing an affine hypothesis in an exponential family the ideal procedure is to calculate the exact similar test or an approximation to this based on the conditional distribution given the minimal sufficient statistic under the null hypothesis by contrast to this there is a primitive approach in which the marginal distribution of a test statistic is considered and any nuisance parameter appearing in the test statistic is replaced by an estimate we show here that when using standardized likelihood ratio statistics the primitive procedure is in fact an ideal procedure to order o n 3 2 as an example we consider inference for the mean of a log normal distribution in detail

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

conditional properties of unconditional parametric bootstrap procedures for inference in exponential families higher order inference about a scalar parameter in the presence of nuisance parameters can be achieved by bootstrapping in circumstances where the parameter of interest is a component of the canonical parameter in a full exponential family the optimal test which is approximated is a conditional one based on conditioning on the sufficient statistic for the nuisance parameter a bootstrap procedure that ignores the conditioning is shown to have desirable conditional properties in providing third order relative accuracy in approximation of p values associated with the optimal test in both continuous and discrete models the bootstrap approach is equivalent to third order analytical approaches and is demonstrated in a number of examples to give very accurate approximations even for very small sample sizes

estimating a population of parameter values using bayes and empirical bayes methods abstract in standard bayes and empirical bayes component decision problems estimating inidividual parameters is the primary goal in multiple comparison problems and in comparisons of histograms of estimates however the primary goal is to produce parameter estimates that can be considered as an ensemble for example the histogram of estimates should be a good estimate of the histogram of parameters standard methods of estimating by the posterior expectation do minimize symmetric componentwise losses such as squared error but they produce ensembles of estimates with a sample variance smaller than the posterior expected sample variance for parameters in this article we propose new bayes and empirical bayes estimates that minimize a distance function between the empirical cdf of the estimates and the true parameters these estimators are weighted averages of the prior mean and the data with weight on the data being approximately the square root of that for the posterior expectation we give theoreti

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

maximum likelihood variance components estimation for binary data abstract we consider a class of probit normal models for binary data and describe ml and reml estimation of variance components for that class as well as best prediction for the realized values of the random effects ml estimates are calculated using an em algorithm for complicated models em includes a gibbs step the computations are illustrated through two examples

random effects models with non parametric priors we discuss the performance of non parametric maximum likelihood npml estimators for the distribution of a univariate random effect in the analysis of longitudinal data for continuous data we analyse generated and real data sets and compare the npml method to those that assume a gaussian random effects distribution and to ordinary least squares for binary outcomes we use generated data to study the moderate and large sample performance of the npml compared with a method based on a gaussian random effect distribution in logistic regression we find that estimated fixed effects are compatible for all approaches but that appropriate standard errors for the npml require adjusting the likelihood based standard errors we conclude that the non parametric approach provides an attractive alternative to gaussian based methods though additional evaluations are necessary before it can be recommended for general use

random effects models for combining results from controlled and uncontrolled studies in a meta analysis abstract a random effects model is used to analyze meta analyses containing both controlled and uncontrolled studies a noniterative estimator of the treatment effect is derived using least squares with the between study variance substituted by an empirical bayes estimator the estimator is shown to be strongly consistent when the between study variance is known and the variance estimator is also shown to be strongly consistent the estimator is shown to have desirable heuristic properties

general empirical bayes wavelet methods and exactly adaptive minimax estimation in many statistical problems stochastic signals can be represented as a sequence of noisy wavelet coefficients in this paper we develop general empirical bayes methods for the estimation of true signal our estimators approximate certain oracle separable rules and achieve adaptation to ideal risks and exact minimax risks in broad collections of classes of signals in particular our estimators are uniformly adaptive to the minimum risk of separable estimators and the exact minimax risks simultaneously in besov balls of all smoothness and shape indices and they are uniformly superefficient in convergence rates in all compact sets in besov spaces with a finite secondary shape parameter furthermore in classes nested between besov balls of the same smoothness index our estimators dominate threshold and james stein estimators within an infinitesimal fraction of the minimax risks more general block empirical bayes estimators are developed both white noise with drift and nonparametric regression are considered

robust empirical bayes estimation of means from stratified samples abstract this article considers simultaneous estimation of means from several strata where each stratum contains a finite number of elements for example quite often estimates of annual incomes unemployment rates and so forth must be made simultaneously for many areas our results find application in small area estimation where very few samples are available from an individual area and an estimate of a certain area mean or simultaneous estimates of several area means can be improved by incorporating information from similar neighboring areas ghosh and meeden 1986 considered empirical bayes estimation of the finite population mean assuming a normal superpopulation model their estimators generalize automatically when one estimates a vector of stratum means this article relaxes the normality assumption and assumes instead that the posterior expectation of any stratum mean is a linear function of the sample observations this is the so called posterior linearity property as described in ericson 1

properties of sufficiency and statistical tests 1 in a previous paper dealing with the importance of properties of sufficiency in the statistical theory of small samples attention was mainly confined to the theory of estimation in the present paper the structure of small sample tests whether these are related to problems of estimation and fiducial distributions or are of the nature of tests of goodness of fit is considered further

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

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

a general framework for model based statistics summary this paper presents a general framework for model based statistics the framework is based on extended concepts of sufficiency and ancillarity the fundamental theorem of the new theory exhibits in a single equation the reference distributions for decision theory model checking and conditional inference for a general statistical model the theorem includes the equations upon which bayesian and frequentist statistics are founded the general theory also fills two gaps previously missing from empirical bayesian statistics concepts of sufficiency and conditional inference section 2 discusses a simple example in depth this example is just complicated enough to illustrate most of the new ideas it allows comparison of the new concepts with those already in the literature because it is so simple essential principles are emphasized section 3 describes the general framework extended concepts of sufficiency and ancillarity form the basis for the framework these extended concepts culminate in a simple but fundamental theorem the generality of the framework is illustrated throughout this section in particular frequentist bayesian robust bayesian empirical bayesian and prediction models are discussed the common framework emphasizes the similarities among various schools of statistical thought in fact besides varying interpreta tions of probability the only essential difference between schools is in the model itself section 4 derives new conditional empirical bayesian results for stein s estimator the estimated conditional mean squared error is shown to equal the posterior mean squared error with respect to the hyperprior that gives rise to stein s estimator section 5 discusses several more complicated empirical bayesian models some sugges tions are offered but for the most part the primary purpose of this section is to indicate the difficulties that must be faced for these problems section 6 offers concluding remarks these remarks summarize the advantages of the general framework

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

standard errors of prediction in generalized linear mixed models abstract the unconditional mean squared error of prediction umsep is widely used as a measure of prediction variance for inferences concerning linear combinations of fixed and random effects in the classical normal theory mixed model but the umsep is inappropriate for generalized linear mixed models where the conditional variance of the random effects depends on the data when the random effects describe variation between independent small domains and domain specific prediction is of interest we propose a conditional mean squared error of prediction cmsep as a general measure of prediction variance the cmsep is shown to be the sum of the conditional variance and a positive correction that accounts for the sampling variability of parameter estimates we derive a second order correct estimate of the cmsep that consists of three components a a plug in estimate of the conditional variance b a plug in estimate of a taylor series approximation to the correction term and c a bootstrap estimate of

approximate bayesian inference with the weighted likelihood bootstrap we introduce the weighted likelihood bootstrap wlb as a way to simulate approximately from a posterior distribution this method is often easy to implement requiring only an algorithm for calculating the maximum likelihood estimator such as iteratively reweighted least squares in the generic weighting scheme the wlb is first order correct under quite general conditions inaccuracies can be removed by using the wlb as a source of samples in the sampling importance resampling sir algorithm which also allows incorporation of particular prior information the sir adjusted wlb can be a competitive alternative to other integration methods in certain models asymptotic expansions elucidate the second order properties of the wlb which is a generalization of rubin s bayesian bootstrap the calculation of approximate bayes factors for model comparison is also considered we note that given a sample simulated from the posterior distribution the required marginal likelihood may be simulation consistently estimated by the harmonic mean of the associated likelihood values a modification of this estimator that avoids instability is also noted these methods provide simple ways of calculating approximate bayes factors and posterior model probabilities for a very wide class of models

simple modifications for signed roots of likelihood ratio statistics summary in the context of inference about a scalar parameter in the presence of nuisance parameters some simple modifications for the signed root of the log likelihood ratio statistic r are developed that reduce the order of error in the standard normal approximation to the distribution of r from o n 1 2 to o n 1 barndorff nielsen has introduced a variable u such that the error in the standard normal approximation to the distribution of r r r log u r is of order o n 3 2 but calculation of urequires the specification of an exact or approximate ancillary statistic a this paper proposes an alternative variable to u denoted by t that is available without knowledge of a and satisfies t u op n 1 in general thus the standard normal approximation to the distribution of r r log t r has error of order o n 1 and it can be used to construct approximate confidence limits having coverage error of order o n 1 in certain cases however tand uare identical the derivation of tinvolves the bayesian approach to constructing confidence limits considered by welch and peers and peers similar modifications for the signed root of the conditional likelihood ratio statistic are also developed and these modifications are seen to be useful when a large number of nuisance parameters are present several examples are presented including inference for natural parameters in exponential models and inference about location scale models with type ii censoring in each case the relationship between tand u is discussed numerical examples are also given including inference for regression models inference about the means of log normal distributions and inference for exponential lifetime models with type i censoring where barndorff nielsen s variable u is not available

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

mean squared error properties of generalized ridge estimators abstract mean squared errors are derived and examined for a number of generalized ridge gr regression estimators and the estimators are compared in a number of specific regression models the potential of gr estimators is discussed

residual diagnostics for mixture models abstract a sample is commonly modeled by a mixture distribution if the observations follow a common distribution but the parameter of interest differs between observations for example we observe the lengths but not the ages of a sample offish it may be reasonable to assume that length is normally distributed about an unknown mean that depends on the age of the fish provided there is more than one age class in the sample then the data are distributed as a mixture of normals in this article we assume that the data are a random sample from a mixture of exponential family distributions and that for each observation the parameter of interest is sampled independently from an unknown mixing distribution q the adequacy of a fitted mixture model can be assessed by examining residuals based on the ratio of the observed to expected fit residuals based on the homogeneity model in which q is a one point distribution display a convexity property when the data follow a mixture model this becomes the basis fo

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

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

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

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

bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator for rather general parametric models a simple connection is established between the bartlett adjustment factor of the log likelihood ratio statistic and the normalizing constant c of the formula c i i 1 2l for the conditional distribution of a maximum likelihood estimator as applied to the full model and the model of the hypothesis tested this leads to a relatively simple demonstration that division of the likelihood ratio statistic by a suitable constant or estimated factor improves the chi squared approximation to its distribution various expressions for these quantities are discussed in particular for the case of a one dimensional parameter an approximation to the constants involved is derived which does not require integration over the sample space

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

empirical bayes confidence intervals based on bootstrap samples abstract consider the model with data generated by the following two stage process first a parameter is sampled from a prior distribution g and then an observation is sampled from the conditional distribution f y if the prior distribution is known then the bayes estimate under squared error loss is the posterior expectation of conditional on the data y for example if g is gaussian with mean and variance 2 and f y is gaussian with mean and variance 2 then the posterior distribution is gaussian with mean b 1 b y and variance 2 1 b where b 2 2 2 inferences about are based on this distribution we study the application of the bootstrap to situations where the prior must be estimated from the data empirical bayes methods for this model we observe data y t y 1 y k t each independent y k following the compound model described previously as first shown by james and stein 1961 setting each k equal to its estimated posterior mean where and pr

effective degrees of freedom and the likelihood ratio test summary normal theory problems involving several unknown variances may require use of estimates not having simple chi squared distributions confidence intervals are then commonly based on the use of effective degrees of freedom i e the fitting of a gamma distribution via the first two moments replacing parameters by estimates the confidence intervals given by this method are compared with those resulting from the likelihood ratio test inserting a bartlett adjustment factor to achieve close approximation by a chi squared distribution two special problems one concerning components of variance and one a generalization of the behrens fisher problem are studied in detail

bayesian inference in econometric models using monte carlo integration methods for the systematic application of monte carlo integration with importance sampling to bayesian inference in econometric models are developed conditions under which the numerical approximation of a posterior moment converges almost surely to the true value as the number of monte carlo replications increases and the numerical accuracy of this approximation may be assessed reliably are set forth methods for the analytical verification of these conditions are discussed

empirical bayes estimation for the finite population mean on the current occasion abstract many finite populations which are sampled repeatedly change slowly over time then estimation of finite population characteristics for the current occasion l may be improved by the use of data from previous surveys in this article we investigate the use of empirical bayes procedures based on two superpopulation models each model has the same first stage the values of the population units on the ith occasion are a random sample from the normal distribution with mean i and variance 2 i at the second stage we assume that either a 1 l are a random sample from the normal distribution with mean and variance 2 or b given 2 i and i has the normal distribution with mean and variance 2 i independently for each i whereas the 2 i are a random sample from the inverse gamma distribution with parameters 2 and 2 in a the 2 i and 2 are assumed to be unknown whereas in b and are unknown we develop empirical bayes point estimators and confide

approximate bayesian inference in conditionally independent hierarchical models parametric empirical bayes models abstract we consider two stage models of the kind used in parametric empirical bayes peb methodology calling them conditionally independent hierarchical models we suppose that there are k units which may be experimental subjects cities study centers etcetera at the first stage the observation vectors yi for units i 1 k are independently distributed with densities p yi i or more generally p yi i at the second stage the unit specific parameter vectors i are iid with densities p i the peb approach proceeds by regarding the second stage distribution as a prior and noting that if were known inference about could be based on its posterior since is not known the simplest peb methods estimate the parameter by maximum likelihood or some variant and then treat as if it were known to be equal to this estimate although this procedure is sometimes satisfactory a well known defect is that it neglects the uncertainty due to the estimation of in this article w

bayesianly justifiable and relevant frequency calculations for the applied statistician a common reaction among applied statisticians is that the bayesian statistician s energies in an applied problem must be directed at the a priori elicitation of one model specification from which an optimal design and all inferences follow automatically by applying bayes s theorem to calculate conditional distributions of unknowns given knowns i feel however that the applied bayesian statistician s tool kit should be more extensive and include tools that may be usefully labeled frequency calculations three types of bayesianly justifiable and relevant frequency calculations are presented using examples to convey their use for the applied statistician

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

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

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 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

the optimal confidence region for a random parameter suppose that under a two level hierarchical model the distribution of the vector of random parameters is known or can be estimated well the data are generated via a fixed but unobservable realisation of the vector we derive the smallest confidence region for a specific component of this random vector under a joint bayesian frequentist paradigm on average this optimal region can be much smaller than the corresponding bayesian highest posterior density region the new estimation procedure is especially appealing when one deals with data generated under a highly parallel structure the new proposal is illustrated with a dataset from a multi centre clinical study and also with one from a typical microarray experiment the performance of our procedure is examined via simulation studies

on parameter transformations and interval estimation summary parameterizations which reduce the asymptotic bias and skewness of various pivotal quantities arising in large sample theory are discussed for models depending on an unknown scalar parameter transformation formulae by which such parameterizations can be obtained are derived and these formulae extend those for one dimensional curved exponential families given by hougaard 1982 to assess the accuracy of normal approximations to the distributions of the pivots their second order properties are considered and comparisons with the signed square root of the likelihood ratio statistic are drawn

a large deviation type approximation for the box class of likelihood ratio criteria abstract for a large class of tests related to the multivariate normal distribution a number of approximations to the distribution of the log likelihood ratio statistic are discussed some of the approximations require the use of a small personal computer for their calculation but this pays off in terms of giving very accurate approximations actually one of the approximations based on the use of a conjugate distribution works well for all sample sizes and all significance levels

on the approximate elimination of nuisance parameters by conditioning summary the general problem of inference about a scalar parameter of interest 0 in the presence of a nuisance parameter a using conditional inference is considered a condition is given under which inference based on the conditional distribution of 0 the maximum likelihood estimate of 0 given l o the maximum likelihood estimate of a for fixed 0 o0 is optimal in a certain sense when this condition is not satisfied it is shown that inference should a aa hee0dntsteosrd be based on the conditional distribution of 0 given lo j where j denotes the observed information for a for fixed 0 00 although this will involve some loss of information about 0 this information loss is shown to be related to the statistical curvature of the model

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 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

detection of significant disease risks using a spatial conditional autoregressive model summary the conditional autoregressive car model is widely used to describe the geographical distribution of a specific disease risk in lattice mapping successful developments based on frequentist and bayesian procedures have been extensively applied to obtain two stage disease risk predictions at the subregional level bayesian procedures are preferred for making inferences as the posterior standard errors se of the two stage prediction account for the variability in the variance component estimates however some recent work based on frequentist procedures and the use of bootstrap adjustments for the se has been undertaken in this article we investigate the suitability of an analytical adjustment for disease risk inference that provides accurate interval predictions by using the penalized quasilikelihood pql technique to obtain model parameter estimates the method is a first order approximation of the naive se based on a taylor expansion and is interpreted as a conditional measure of variability providing conditional calibrated prediction intervals given the data we conduct a simulation study to demonstrate how the method can be used to estimate the specific subregion risk by interval we evaluate the proposed methodology by analyzing the commonly used example data set of lip cancer incidence in the 56 counties of scotland for the period 1975 1980 this evaluation reveals a close similarity between the solutions provided by the method proposed here and those of its fully bayesian counterpart