brunn minkowski inequality for multiplicities this is a convex subset of the real vector space p zr in fact it is a convex polytope see br where this polytope is discussed from an algebraic point of view it is known that the total mass of m is a polynomial in m for su ciently large m denote by k its degree and m k m m weak d where d is the lebesgue measure supported on and the density is a piecewise polynomial function we will not use this piecewise polynomiality in this paper recall that a real function f de ned on a convex subset u of a vector space v is called concave if

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

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

analysis of multi type recurrent events in longitudinal studies application to a skin cancer prevention trial we consider the statistical modeling and analysis of replicated multi type point process data with covariates such data arise when heterogeneous subjects experience repeated events or failures which may be of several distinct types the underlying processes are modeled as nonhomogeneous mixed poisson processes with random subject and fixed covariate effects the method of maximum likelihood is used to obtain estimates and standard errors of the failure rate parameters and regression coefficients score tests and likelihood ratio statistics are used for covariate selection a graphical test of goodness of fit of the selected model is based on generalized residuals measures for determining the influence of an individual observation on the estimated regression coefficients and on the score test statistic are developed an application is described to a large ongoing randomized controlled clinical trial for the efficacy of nutritional supplements of selenium for the prevention of two types of skin cancer

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

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

theoretical explanation of observed decreasing failure rate pooled data on the times of successive failures of the air conditioning system of a fleet of jet airplanes seemed to indicate that the life distribution had a decreasing failure rate more refined analysis showed that the failure distribution for each airplane separately was exponential but with a different failure rate using the theorem that a mixture of distributions each having a non increasing failure rate e g a mixture of exponential distributions itself has a non increasing failure rate the apparent decreasing failure rate of the pooled air conditioning life distribution was satisfactorily explained the present study has implications in other areas where an observed decreasing failure rate may well be the result of mixing exponential distributions having different parameters

a monte carlo implementation of the em algorithm and the poor man s data augmentation algorithms abstract the first part of this article presents the monte carlo implementation of the e step of the em algorithm given the current guess to the maximizer of the posterior distribution latent data patterns are generated from the conditional predictive distribution the expected value of the augmented log posterior is then updated as a mixture of augmented log posteriors mixed over the generated latent data patterns multiple imputations in the m step of the algorithm this mixture is maximized to obtain the update to the maximizer of the observed posterior the gradient and hessian of the observed log posterior are also expressed as mixtures mixed over the multiple imputations the relation between the monte carlo em mcem algorithm and the data augmentation algorithm is noted two modifications to the mcem algorithm the poor man s data augmentation algorithms which allow for the calculation of the entire posterior are then presented these approximations serve as diagnostics for the validity o

fixed width output analysis for markov chain monte carlo markov chain monte carlo is a method of producing a correlated sample to estimate features of a target distribution through ergodic averages a fundamental question is when sampling should stop that is at what point the ergodic averages are good estimates of the desired quantities we consider a method that stops the simulation when the width of a confidence interval based on an ergodic average is less than a user specified value hence calculating a monte carlo standard error is a critical step in assessing the simulation output we consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution we give sufficient conditions for the strong consistency of both methods and investigate their finite sample properties in various examples

hierarchical generalized linear models we consider hierarchical generalized linear models which allow extra error components in the linear predictors of generalized linear models the distribution of these components is not restricted to be normal this allows a broader class of models which includes generalized linear mixed models we use a generalization of henderson s joint likelihood called a hierarchical or h likelihood for inferences from hierarchical generalized linear models this avoids the integration that is necessary when marginal likelihood is used under appropriate conditions maximizing the h likelihood gives fixed effect estimators that are asymptotically equivalent to those obtained from the use of marginal likelihood at the same time we obtain the random effect estimates that are asymptotically best unbiased predictors an adjusted profile h likelihood is shown to give the required generalization of restricted maximum likelihood for the estimation of dispersion components a scaled deviance test for the goodness of fit a model selection criterion for choosing between various dispersion models and a graphical method for checking the distributional assumption of random effects are proposed the ideas of quasi likelihood and extended quasi likelihood are generalized to the new class we give examples of the poisson gamma binomial beta and gamma inverse gamma hierarchical generalized linear models a resolution is proposed for the apparent difference between population averaged and subject specific models a unified framework is provided for viewing and extending many existing methods

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

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

shape topology optimization for navier stokes problem using variational level set method we consider the shape topology optimization of the navier stokes problem a new algorithm is proposed based on the variational level set method by this algorithm a relatively smooth evolution can be maintained without re initialization and drastic topology change can be handled easily finally the promising features of the proposed method are illustrated by two benchmark examples

discretization of continuous markov chains and markov chain monte carlo convergence assessment abstract we show that continuous state space markov chains can be rigorously discretized into finite markov chains the idea is to subsample the continuous chain at renewal times related to small sets that control the discretization once a finite markov chain is derived from the markov chain monte carlo output general convergence properties on finite state spaces can be exploited for convergence assessment in several directions our choice is based on a divergence criterion derived from kemeny and snell which is first evaluated on parallel chains with a stopping time and then implemented more efficiently on two parallel chains only using birkhoff s pointwise ergodic theorem for stopping rules the performance of this criterion is illustrated on three standard examples

the gamma frailty poisson model for the nonparametric estimation of panel count data summary in this article we study nonparametric estimation of the mean function of a counting process with panel observations we introduce the gamma frailty variable to account for the intracorrelation between the panel counts of the counting process and construct a maximum pseudo likelihood estimate with the frailty variable three simulated examples are given to show that this estimation procedure while preserving the robustness and simplicity of the computation improves the efficiency of the nonparametric maximum pseudo likelihood estimate studied in wellner and zhang 2000 annals of statistics28 779 814 a real example from a bladder tumor study is used to illustrate the method

minorization conditions and convergence rates for markov chain monte carlo abstract general methods are provided for analyzing the convergence of discrete time general state space markov chains such as those used in stochastic simulation algorithms including the gibbs sampler the methods provide rigorous a priori bounds on how long these simulations should be run to give satisfactory results results are applied to two models of the gibbs sampler a bivariate normal model and a hierarchical poisson model with gamma conditionals the methods use the notion of minorization conditions for markov chains

the application of domain derivative for heat conduction with mixed condition in shape reconstruction we consider the inverse problem for heat conduction with mixed condition to reconstruct the shape of multiple connected domain a representation for domain derivative of the corresponding operator is established this allows the investigation of the regularized iterative method for the solution of this ill posed and nonlinear problem the results of several numerical experiments show that the iterative algorithm gives good reconstruction and indicate the feasibility of this method

an efficient sampling technique for bayesian inference with computationally demanding models in the environmental sciences a large knowledge base is typically available on an investigated system or at least on similar systems this makes the application of bayesian inference techniques in environmental modeling very promising however environmental systems are often described by complex computationally demanding simulation models this strongly limits the application of bayesian inference techniques because numerical implementation of these techniques requires a very large number of simulation runs the development of efficient sampling techniques that attempt to approximate the posterior distribution with a relatively small parameter sample can extend the range of applicability of bayesian inference techniques to such models in this article a sampling technique is presented that tries to achieve this goal the proposed technique combines numerical techniques typically applied in bayesian inference including posterior maximization local normal approximation and importance sampling with c

a technique for estimating marginal posterior densities in hierarchical models using mixtures of conditional densities abstract a technique called quantile integration is proposed for the estimation of marginal posterior densities arising in bayesian models having hierarchical representations the method is based on approximating marginal densities as mixtures of conditional densities where the conditioning variables are selected deterministically from the mixing distributions the form of the approximation makes it easy to implement and the resulting approximations are computationally efficient to obtain the technique leads to particularly simple approximations for the predictive and posterior densities in kalman filter or state space models and specific formulas are provided for the special case in which innovations belong to location scale families other applications include a hierarchical empirical bayes model for poisson rates and a hierarchical linear model with exchangeable regression parameters and unknown variance components

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

okounkov bodies and toric degenerations let vardelta be the okounkov body of a divisor d on a projective variety x we describe a geometric criterion for vardelta to be a lattice polytope and show that in this situation x admits a flat degeneration to the corresponding toric variety this degeneration is functorial in an appropriate sense

estimation of trends in the scram rate at nuclear power plants an important task of the u s nuclear regulatory commission is to examine annual operating data from the nation s population of nuclear power plants for trends over time we are interested here in trends in the scram rate at 66 commercial nuclear power plants based on annual observed scram data from 1984 1993 for an assumed poisson distribution on the number of unplanned scrams a gamma prior and an appropriate hyperprior a parametric empirical bayes peb approximation to a full hierarchical bayes formulation is used to estimate the scram rate for each plant for each year the peb estimated prior and posterior distributions are then subsequently smoothed over time using an exponentially weighted moving average the results indicate that such bidirectional shrinkage is quite useful for identifying reliability trends over time

nonparametric k sample tests with panel count data we study the nonparametric k sample test problem with panel count data the asymptotic normality of a smooth functional of the nonparametric maximum pseudolikelihood estimator wellner zhang 2000 is established under some mild conditions we construct a class of easy to implement nonparametric tests for comparing mean functions of k populations based on this asymptotic normality we conduct various simulations to validate and compare the tests the simulations show that the tests perform quite well and generally have good power to detect differences among the mean functions the method is illustrated with a real life example

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

derandomizing variance estimators one may consider a discrete event simulation as a markov chain evolving on a suitably rich state space one way that regenerative cycles may be constructed for general state space markov chains is to generate auxiliary coin flip random variables at each transition with a regeneration occurring if the coin flip results in a success the regenerative cycles are therefore randomized with respect to the sequence of states visited by the markov chain the point estimator for a steady state performance measure does not depend on the cycle structure of the chain but the variance estimator that defines the width of a confidence interval for the performance measure does this implies that the variance estimator is randomized with respect to the visited states we show how to derandomize the variance estimator through the use of conditioning a new variance estimator is obtained that is consistent and has lower variance than the standard estimator

shape reconstruction of an inverse stokes problem this paper deals with the shape reconstruction of a viscous incompressible fluid driven by the stokes flow for the approximate solution of the ill posed and nonlinear problem we propose a regularized newton method a theoretical foundation for the newton method is given by establishing the differentiability of the initial boundary value problem with respect to the interior boundary curve in the sense of a domain derivative the numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible

okounkov bodies on projectivizations of rank two toric vector bundles abstract the global okounkov body of a projective variety is a closed convex cone that encodes asymptotic information about every big line bundle on the variety in the case of a rank two toric vector bundle e on a smooth projective toric variety we use its klyachko filtrations to give an explicit description of the global okounkov body of p e in particular we show that this is a rational polyhedral cone and that p e is a mori dream space

two likelihood based semiparametric estimation methods for panel count data with covariates we consider estimation in a particular semiparametric regression model for the mean of a counting process with panel count data the basic model assumption is that the conditional mean function of the counting process is of the form efn t jzg exp fl t z 0 t where z is a vector of covariates and 0 is the baseline mean function the panel count observation scheme involves observation of the counting processn for an individual at a random number k of random time points both the number and the locations of these time points may difier across individuals we study semiparametric maximum pseudo likelihood and maximum likelihood estimators of the unknown parameters fl0 0 derived on the basis of a nonhomogeneous poisson process assumption the pseudo likelihood estimator is fairly easy to compute while the maximum likelihood estimator poses more challenges from the computational perspective we study asymptotic properties of both estimators assuming that the proportional mean model holds but dropping the poisson process assumption used to derive the estimators in particular we establish asymptotic normality for the estimators of the regression parameter fl0 under appropriate hypotheses the results show that our estimation procedures are robust in the sense that the estimators converge to the truth regardless of the underlying counting process

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

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

semi nonparametric maximum likelihood estimation on developpe une approche basee sur une innovation due a phillips 1983 sur les approximations des fonctions de densite

testing for monotone trend in recurrent event processes tests for the presence of monotone trends in recurrent event processes have been proposed and studied by many authors however a general concept of trend is elusive and the dependence of the behavior of tests on the assumed definitions of no trend and trend and on the observation periods for the processes has been largely ignored we discuss these issues and study them through analytical results and simulation studies the article also presents robust tests for trend across multiple processes extends them to deal with interval censored event times and compares them and other well known trend tests with respect to the issues mentioned

robustification of kalman filter models abstract kalman filter models based on the assumption of multivariate gaussian distributions are known to be nonrobust this means that when a large discrepancy arises between the prior distribution and the observed data the posterior distribution becomes an unrealistic compromise between the two in this article we discuss a rationale for how to robustify the kalman filter specifically we develop a model wherein the posterior distribution will revert to the prior when extreme outlying observations are encountered and we point out that this can be achieved by assuming a multivariate distribution with student t marginals to achieve fully robust results of the kind desired it becomes necessary to forsake an exact distribution theory approach and adopt an approximation method involving poly t distributions a recursive mechanism for implementing the multivariate t based kalman filter is described its properties are discussed and the procedure is illustrated by an example

regeneration in markov chain samplers abstract markov chain sampling has recently received considerable attention in particular in the context of bayesian computation and maximum likelihood estimation this article discusses the use of markov chain splitting originally developed for the theoretical analysis of general state space markov chains to introduce regeneration into markov chain samplers this allows the use of regenerative methods for analyzing the output of these samplers and can provide a useful diagnostic of sampler performance the approach is applied to several samplers including certain metropolis samplers that can be used on their own or in hybrid samplers and is illustrated in several examples

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea