harmonic maass jacobi forms with singularities and a theta like decomposition real analytic jacobi forms play key roles in different areas of mathematics and physics but a satisfactory theory of such jacobi forms has been lacking in this paper we fill this gap by introducing a space of harmonic maass jacobi forms with singularities which includes the real analytic jacobi forms from zwegers s phd thesis we provide several structure results for the space of such jacobi forms and we employ zwegers s functions to establish a theta like decomposition

the estimation of the mean squared error of small area estimators abstract small area estimation has received considerable attention in recent years because of a growing demand for reliable small area statistics the direct survey estimators based only on the data from a given small area or small domain are likely to yield unacceptably large standard errors because of small sample size in the domain therefore alternative estimators that borrow strength from other related small areas have been proposed in the literature to improve the efficiency these estimators use models either implicitly or explicitly that connect the small areas through supplementary e g census and administrative data for example simple synthetic estimators are based on implicit modeling in this article three small area models of battese harter and fuller 1988 dempster rubin and tsutakawa 1981 and fay and herriot 1979 are investigated these models are all special cases of a general mixed linear model involving fixed and random effects and a small area mean can be expr

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

kohnen s limit process for real analytic siegel modular forms abstract kohnen introduced a limit process for siegel modular forms that produces jacobi forms he asked if there is a space of real analytic siegel modular forms such that skew holomorphic jacobi forms arise via this limit process in this paper we initiate the study of harmonic skew maass jacobi forms and harmonic siegel maass forms we improve a result of maass on the fourier coefficients of harmonic siegel maass forms which allows us to establish a connection to harmonic skew maass jacobi forms in particular we answer kohnen s question in the affirmative

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

posterior distribution of hierarchical models using car 1 distributions summary we examine properties of the conditional autoregressive model or car 1 model which is commonly used to represent regional effects in bayesian analyses of mortality rates we consider a bayesian hierarchical linear mixed model where the fixed effects have a vague prior such as a constant prior and the random effect follows a class of car 1 models including those whose joint prior distribution of the regional effects is improper we give sufficient conditions for the existence of the posterior distribution of the fixed and random effects and variance components we then prove the necessity of the conditions and give a one way analysis of variance example where the posterior may or may not exist finally we extend the result to the generalised linear mixed model which includes as a special case the poisson log linear model commonly used in disease mapping

semiparametric inference in generalized mixed effects models summary the paper presents a study of the generalized partially linear model including random effects in its linear part we propose an estimator that combines likelihood approaches for mixed effects models with kernel methods following the methodology of hardie and co workers we introduce a test for the hypothesis of a parametric mixed effects model against the alternative of a semiparametric mixed effects model the critical values are estimated by using a bootstrap procedure the asymptotic theory for the methods is provided as are the results of a simulation study these verify the feasibility and the excellent behaviour of the methods for samples of even moderate size the usefulness of the methodology is illustrated with an application in which the objective is to estimate forest coverage in galicia spain

twisting with a flip the art of pestunization we construct mathcal n 2 supersymmetric yang mills theory on 4d manifolds with a killing vector field with isolated fixed points it turns out that for every fixed point one can allocate either instanton or anti instanton contributions to the partition function and that this is compatible with supersymmetry the equivariant donaldson witten theory is a special case of our construction we present a unified treatment of pestun s calculation on s 4 and equivariant donaldson witten theory by generalizing the notion of self duality on manifolds with a vector field we conjecture the full partition function for a mathcal n 2 theory on any 4d manifold with a killing vector using this new notion of self duality to localize a supersymmetric theory is what we call pestunization

a bayesian space varying parameter model applied to estimating fertility schedules we propose a spatial generalized linear model glm to analyse the vital rates for small areas in each small area we have a response vector and covariates to explain its variability the statistical methodology is based on a spatial bayesian approach and it allows the covariates parameters of the generalized linear model to vary smoothly on space hence the effect of a covariate on the response varies depending on the random variables measurement location our model is an extension of disease mapping models allowing the space covariate interaction to be modelled in a natural way and giving space a position of intrinsic interest we introduce the model in the context of fertility curve estimation in each small area we have a curve describing the variation of fertility rates by age modelled by coale s fertility model which implies a glm in each area a simulation shows the advantages of our approach in addition the paper applies the procedure to census data used to study the diffusion of low fertility behaviour in brazil

bayesian analysis of space time variation in disease risk the analysis of variation of risk for a given disease in space and time is a key issue in descriptive epidemiology when the data are scarce maximum likelihood estimates of the area specific risk and of its linear time trend can be seriously affected by random variation in this paper we propose a bayesian model in which both area specific intercept and trend are modelled as random effects and correlation between them is allowed for this model is an extension of that originally proposed for disease mapping it is illustrated by the analysis of the cumulative prevalence of insulin dependent diabetes mellitus as observed at the military examination of 18 year old conscripts born in sardinia during the period 1936 1971 data concerning the genetic differentiation of the sardinian population are used to interpret the results

identifiability and convergence issues for markov chain monte carlo fitting of spatial models the marked increase in popularity of bayesian methods in statistical practice over the last decade owes much to the simultaneous development of markov chain monte carlo mcmc methods for the evaluation of requisite posterior distributions however along with this increase in computing power has come the temptation to fit models larger than the data can readily support meaning that often the propriety of the posterior distributions for certain parameters depends on the propriety of the associated prior distributions an important example arises in spatial modelling wherein separate random effects for capturing unstructured heterogeneity and spatial clustering are of substantive interest even though only their sum is well identified by the data increasing the informative content of the associated prior distributions offers an obvious remedy but one that hampers parameter interpretability and may also significantly slow the convergence of the mcmc algorithm in this paper we investigate the relationship among identifiability bayesian learning and mcmc convergence rates for a common class of spatial models in order to provide guidance for prior selection and algorithm tuning we are able to elucidate the key issues with relatively simple examples and also illustrate the varying impacts of covariates outliers and algorithm starting values on the resulting algorithms and posterior distributions

estimation of cancer mortality rates a bayesian analysis of small frequencies a bayesian method is presented for estimating mortality rates of specific diseases when the frequency of deaths over a specified time period is assumed to have a poisson distribution with mean proportional to the population size the estimators use information from related populations each having its own rate which is assumed distributed according to a common prior distribution about which some information is available the method is illustrated using data from an epidemiological study on the geographic variation of cancer mortality in missouri comparisons are also made with a simpler empirical bayes method excerpt

hierarchical bayes estimation of hunting success rates with spatial correlations a bayesian hierarchical generalized linear model is used to estimate hunting success rates at the subarea level for postseason harvest surveys the model includes fixed week effects random geographic effects and spatial correlations between neighboring subareas the computation is done by gibbs sampling and adaptive rejection sampling techniques the method is illustrated using data from the missouri turkey hunting survey in the spring of 1996 bayesian model selection methods are used to demonstrate that there are significant week differences and spatial correlations of hunting success rates among counties the bayesian estimates are also shown to be quite robust in terms of changes of hyperparameters

spatial modeling of regional variables abstract in this article accumulated sudden infant death syndrome sids data from 1974 1978 and 1979 1984 for the counties of north carolina are analyzed after a spatial exploratory data analysis markov random field models are fit to the data the spatial trend is meant to capture the large scale variation in the data and the variance and spatial dependence are meant to capture the small scale variation the trend could be a function of other explanatory variables or could simply be modeled as a function of spatial location both models are fit and compared the results give an excellent illustration of a phenomenon already well known in time series that autocorrelation in data can be due to an undiscovered explanatory variable indeed for 1974 1978 we confirm a dependence of sids rate on proportion of nonwhite babies born along with insignificant spatial correlation without this regressor variable however the spatial correlation is significant in 1979 1984 perhaps due to reporting bias o

the effect of improper priors on gibbs sampling in hierarchical linear mixed models abstract often either from a lack of prior information or simply for convenience variance components are modeled with improper priors in hierarchical linear mixed models although the posterior distributions for these models are rarely available in closed form the usual conjugate structure of the prior specification allows for painless calculation of the gibbs conditionals thus the gibbs sampler may be used to explore the posterior distribution without ever having established propriety of the posterior an example is given showing that the output from a gibbs chain corresponding to an improper posterior may appear perfectly reasonable thus one cannot expect the gibbs output to provide a red flag informing the user that the posterior is improper the user must demonstrate propriety before a markov chain monte carlo technique is used a theorem is given that classifies improper priors according to the propriety of the resulting posteriors applications concerning bayesian analysis of animal breeding

virtual refinements of the vafa witten formula we conjecture a formula for the generating function of virtual chi y genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2 form specializing the conjecture to minimal surfaces of general type and to virtual euler characteristics we recover part of a formula of c vafa and e witten these virtual chi y genera can be written in terms of descendent donaldson invariants using t mochizuki s formula the latter can be expressed in terms of seiberg witten invariants and certain explicit integrals over hilbert schemes of points these integrals are governed by seven universal functions which are determined by their values on mathbb p 2 and mathbb p 1 times mathbb p 1 using localization we calculate these functions up to some order which allows us to check our conjecture in many cases in an appendix by h nakajima and the first named author the virtual euler characteristic specialization of our conjecture is extended to include mu classes thereby interpolating between vafa witten s formula and witten s conjecture for donaldson invariants

empirical bayes estimates of age standardized relative risks for use in disease mapping there have been many attempts in recent years to map incidence and mortality from diseases such as cancer such maps usually display either relative rates in each district as measured by a standardized mortality ratio smr or some similar index or the statistical significance level for a test of the difference between the rates in that district and elsewhere neither of these approaches is fully satisfactory and we propose a new approach using empirical bayes estimation the resulting estimators represent a weighted compromise between the smr the overall mean relative rate and a local mean of the relative rate in nearby areas the compromise solution depends on the reliability of each individual smr and on estimates of the overall amount of dispersion of relative rates over different districts

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

the nekrasov conjecture for toric surfaces the nekrasov conjecture predicts a relation between the partition function for n 2 supersymmetric yang mills theory and the seiberg witten prepotential for instantons on mathbb r 4 the conjecture was proved independently and using different methods by nekrasov okounkov and nakajima yoshioka we prove a generalized version of the conjecture for instantons on noncompact toric surfaces

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

bayesian predictive inference for a finite population proportion two stage cluster sampling summary given binary data from a two stage cluster sample we present a method to carry out bayesian predictive inference for a finite population proportion our probabilistic specification should be useful for many surveys of this type and yields simple analytical expressions for the prior and posterior mean and variance within cluster k we assume that the yki are a random sample from the bernoulli distribution with probability ok conditional on b and t 01 on are a random sample from a beta distribution finally 13 has a discrete distribution with specified probabilities we use data from the national health interview survey to illustrate the methodology and to show how to choose values for the parameters in the prior distribution

instanton counting on blowup i 4 dimensional pure gauge theory we give a mathematically rigorous proof of nekrasov s conjecture the integration in the equivariant cohomology over the moduli spaces of instantons on 4 gives a deformation of the seiberg witten prepotential for n 2 susy yang mills theory through a study of moduli spaces on the blowup of 4 we derive a differential equation for the nekrasov s partition function it is a deformation of the equation for the seiberg witten prepotential found by losev et al and further studied by gorsky et al

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

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

hierarchical bayesian spatiotemporal analysis of revascularization odds using smoothing splines hierarchical bayesian models are proposed for over dispersed longitudinal spatially correlated binomial data this class of models accounts for correlation among regions by using random effects and allows a flexible modelling of spatiotemporal odds by using smoothing splines the aim is i to develop models which will identify temporal trends of odds and produce smoothed maps including regional effects ii to specify markov chain monte carlo mcmc inference for fitting such models iii to study the sensitivity of such bayesian binomial spline spatiotemporal analyses to prior assumptions and iv to compare mechanisms for assessing goodness of fit an analysis of regional variation for revascularization odds of patients hospitalized for acute coronary syndrome in quebec motivates and illustrates the methods developed

empirical bayes estimation in finite population sampling abstract empirical bayes methods are becoming increasingly popular in statistics robbins 1955 introduced the method in the context of nonparametric estimation of a completely unspecified prior distribution subsequently the method has been explored very successfully in a series of articles by efron and morris 1973 1975 1977 in a parametric framework in the efron morris setup a family of parametric distributions is used as possible priors but only when one or more of the parameters of the family of prior distributions is estimated from the data morris 1983 listed a number of areas where empirical bayes methods are used one of the main features of empirical bayes analysis is to borrow strength from the ensemble that is use information from similar sources in constructing estimators and predictors in addition to the most directly available source of information there are some situations in finite population sampling where such methods might be suitable for instance in many repetitive surve

quantum yang mills on a riemann surface we obtain the quantum expectations of gauge invariant functions of the connection on ag su n product bundle over a riemann surface of genusg we show that the spacea g m of connections modulo those gauge transformations which are the identity at one point is itself a principal bundle with affine linear fiber the base space path2gg consists of 2g tuples of paths ing subject to a relation on their endpoint values quantum expectations are iterated path integrals over first the fiber then over path2gg each with respect to the push forward toa g m of the measuree s a da here s a denotes the yang mills action ona we exhibit a global section ofa g m to define a choice of origin in each fiber relative to which the measure on the fiber is gaussian the induced measure on path2gg is the product of wiener measures on the component paths conditioned to preserve the endopoint relation conformal transformations of the metric onm act by reparametrizing these paths we explicitly compute the partition function in the general case and the expectations of functions of certain products of wilson loops in the caseg 1

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

the blowup formula for donaldson invariants k k which is calculated in 4 below this formula has been the target of much recent work the abstract fact that there exists such a formula which is independent of x was first proved by c taubes using techniques of t j bryan b and p ozsvath o have independently calculated the coefficients throughb10 x quite recently j morgan and ozsvath have announced a scheme which can recursively compute all of the bk x the special case of the blowup formula for manifolds of simple type see 5 below was first given by p kronheimer and t mrowka however none of the techniques in these cases approach the simplicity of that offered here before presenting the formula we shall first establish notation for the donaldson in variants of a simply connected 4 manifold x with b 1 and odd the hypothesis of simple connectivity is not necessary but makes the exposition easier an orientation of x together with an orientation of h 2 x r is called a homology orientation of x such a homology orientation determines the su 2 donaldson invariant a linear function d dx a x sym h0 x h2 x r

modelling complexity applications of gibbs sampling in medicine summary we review applications of gibbs sampling in medicine involving longitudinal spatial covariate measurement and survival models applications in immunology pharmacology transplantation cancer screening industrial epidemiology and genetic epidemiology are discussed

quantum yang mills on the two sphere we obtain the quantum expectations of gauge invariant functions of the connection on a principalg su n bundle overs 2 we show that the spacea g m of connections modulo gauge transformations which are the identity at one point is itself a principal bundle over g based loops in the symmetry group the fiber ina g m is an affine linear space quantum expectations are iterated path integrals first over this fiber then over g each with respect to the push forward toa g m of the measure s s a da s a denotes the yang mills action ona there is a global section ofa g m on which the first integral is a gaussian the resulting measure on g is the conditional wiener measure we explicitly compute the expectations of a special class of wilson loops

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

energy in yang mills on a riemann surface sengupta s lower bound for the yang mills action on smooth connections on a bundle over a riemann surface generalizes to the space of connections whose action is finite in this larger space the inequality can always be saturated the yang mills critical sets correspond to critical sets of the energy action on a space of paths this may shed light on a conjecture of atiyah and bott concerning morse theory for a g

a bayesian semi parametric model for colorectal cancer incidences a bayesian semi parametric model is proposed to capture the interaction among demographic effects age and gender spatial effects county and temporal effects of colorectal cancer incidences simultaneously in particular an extension of multivariate conditionally autoregressive car processes to a partially informative gaussian demographic spatial temporal car dstcar process for a spatial temporal setting is proposed the precision matrix of the gaussian dstcar process is the kronecker product of several components the spatial component is modelled with a car prior a pth order intrinsic autoregressive prior iar p is implemented for the temporal component to estimate a smoothed and non parametric temporal trend the demographic component is modelled with a wishart prior data analysis shows significant spatial correlation only exists in the age group of 50 59 males and females in their 50s and 60s show fairly strong correlation the hypothesis testing based on bayes factor suggests that gender correlation cannot be ignored in this model

fully bayesian spline smoothing and intrinsic autoregressive priors summary there is a well known bayesian interpretation for function estimation by spline smoothing using a limit of proper normal priors the limiting prior and the conditional and intrinsic autoregressive priors popular for spatial modelling have a common form which we call partially informative normal we derive necessary and sufficient conditions for the propriety of the posterior for this class of partially informative normal priors with noninformative priors on the variance components a condition crucial for successful implementation of the gibbs sampler the results apply for fully bayesian smoothing

model choice a minimum posterior predictive loss approach summary model choice is a fundamental and much discussed activity in the analysis of datasets nonnested hierarchical models introducing random effects may not be handled by classical methods bayesian approaches using predictive distributions can be used though the formal solution which includes bayes factors as a special case can be criticised we propose a predictive criterion where the goal is good prediction of a replicate of the observed data but tempered by fidelity to the observed values we obtain this criterion by minimising posterior loss for a given model and then for models under consideration selecting the one which minimises this criterion for a broad range of losses the criterion emerges as a form partitioned into a goodness of fit term and a penalty term we illustrate its performance with an application to a large dataset involving residential property transactions

computational methods using a bayesian hierarchical generalized linear model abstract suppose we observe y 1 yn where yi has the exponential density f yi i exp i yi i b i c yi i the parameters of interest are not the canonical parameters i but the means i b i in the usual generalized linear model glm setup suppose the means 1 n are believed to satisfy a specific p dimensional glm g i xt i where the link function g and the regression coefficients xi are known and the regression vector is unknown two problems of interest are the assessment of the goodness of fit of the glm and the estimation of the means i the approach to these problems is by the use of a bayesian two stage prior distribution a generalization of a model used by lindley and smith 1972 in the normal mean estimation problem at the first stage of the model we assign independent conjugate distributions to 1 n where the prior means of the i satisfy the glm there are p 1 unknown hyperparameters in this specification the elements of the regression

a bivariate bayes method for improving the estimates of mortality rates with a twofold conditional autoregressive model a bivariate bayes method is proposed for estimating the mortality rates of a single disease for a given population using additional information from a second disease the information on the two diseases is assumed to be from the same population groups or areas the joint frequencies of deaths for the two diseases for given populations are assumed to have a bivariate poisson distribution with joint means proportional to the population sizes the relationship between the mortality rates of the two different diseases if formulated through the twofold conditional autoregressive car model where spatial effects as well as indexes of spatial dependence are introduced to capture the structured clusterings among areas this procedure is compared to a univariate hierarchical bayes procedure that uses information from one disease only comparisons of two procedures are made by the optimal property a monte carlo study real data and the bayes factor all of the methods that we consider demonstrate a substantial

small area estimates for cross classifications we develop a class of log linear structural models that is suited to estimation of small area cross classified counts based on survey data this allows us to account for various association structures within the data and includes as a special case the restricted log linear model underlying structure preserving estimation the effect of survey design can be incorporated into estimation through the specification of an unbiased direct estimator and its associated covariance structure we illustrate our approach by applying it to estimation of small area labour force characteristics in norway

framed sheaves on root stacks and supersymmetric gauge theories on ale spaces we develop a new approach to the study of supersymmetric gauge theories on ale spaces using the theory of framed sheaves on root toric stacks which illuminates relations with gauge theories on r 4 and with two dimensional conformal field theory we construct a stacky compactification of the minimal resolution xk of the ak 1 toric singularity c 2 zk which is a projective toric orbifold xk such that xknxk is a zk gerbe we construct moduli spaces of tor sion free sheaves on xk which are framed along the compactification gerbe we prove that this moduli space is a smooth quasi projective variety compute its dimension and classify its fixed points under the natural induced toric action we use this construction to compute the partition functions and correlators of chiral bps operators forn 2 quiver gauge theories on xk with nontrivial holonomies at infinity the partition functions are computed with and without cou plings to bifundamental matter hypermultiplets and expressed in terms of toric blowup formulas which relate them to the corresponding nekrasov partition functions on the affine toric subsets of xk we compare our new partition functions with previous computations explore their con nections to the representation theory of affine lie algebras and find new constraints on fractional instanton charges in the coupling to fundamental matter we show that the partition functions in the low energy limit are characterised by the seiberg witten curves and in some cases also by suitable blowup equations involving riemann theta functions on the seiberg witten curve with characteristics related to the nontrivial holonomies

extending conventional priors for testing general hypotheses in linear models summary we consider that observations come from a general normal linear model and that it is desirable to test a simplifying null hypothesis about the parameters we approach this problem from an objective bayesian model selection perspective crucial ingredients for this approach are proper objective priors to be used for deriving the bayes factors jeffreys zellner siow priors have good properties for testing null hypotheses defined by specific values of the parameters in full rank linear models we extend these priors to deal with general hypotheses in general linear models not necessarily of full rank the resulting priors which we call conventional priors are expressed as a generalization of recently introduced partially informative distributions the corresponding bayes factors are fully automatic easily computed and very reasonable the methodology is illustrated for the change point problem and the equality of treatments effects problem we compare the conventional priors derived for these problems with other objective bayesian proposals like the intrinsic priors it is concluded that both priors behave similarly although interesting subtle differences arise we adapt the conventional priors to deal with nonnested model selection as well as multiple model comparison finally we briefly address a generalization of conventional priors to nonnormal scenarios

instanton counting and donaldson invariants for a smooth projective toric surface we determine the donaldson invariants and their wallcrossing in terms of the nekrasov partition function using the solution of the nekrasov conjecture 33 38 3 and its refinement 34 we apply this result to give a generating function for the wallcrossing of donaldson invariants of good walls of simply connected projective surfaces with b 1 in terms of modular forms this formula was proved earlier in 19 more generally for simply connected 4 manifolds with b 1 assuming the kotschick morgan conjecture and it was also derived by physical arguments in 31

generalized linear models with random effects a gibbs sampling approach abstract generalized linear models have unified the approach to regression for a wide variety of discrete continuous and censored response variables that can be assumed to be independent across experimental units in applications such as longitudinal studies genetic studies of families and survey sampling observations may be obtained in clusters responses from the same cluster cannot be assumed to be independent with linear models correlation has been effectively modeled by assuming there are cluster specific random effects that derive from an underlying mixing distribution extensions of generalized linear models to include random effects has thus far been hampered by the need for numerical integration to evaluate likelihoods in this article we cast the generalized linear random effects model in a bayesian framework and use a monte carlo method the gibbs sampler to overcome the current computational limitations the resulting algorithm is flexible to easily accommodate changes in the number

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

bayesian factor analysis for spatially correlated data with application to summarizing area level material deprivation from census data this article describes a bayesian hierarchical model for factor analysis of spatially correlated multivariate data the first level specifies for each area on a map the distribution of a vector of manifest variables conditional on an underlying latent factor at the second level the area specific latent factors have a joint distribution that incorporates spatial correlation the framework allows for both marginal and conditional e g conditional autoregressive specifications of spatial correlation the model is used to quantify material deprivation at the census tract level using data from the 1990 u s census in rhode island an existing and widely used measure of material deprivation is the townsend index an unweighted sum of four standardized census variables i e z scores corresponding to area level proportions of unemployment car ownership crowding and home ownership the townsend and many related indices are computed as linear combinations of measured census variables which motivates t

robust estimation in generalised linear mixed models summary we propose robust methods for estimation of parameters of interest in an extended generalised linear mixed model in which only the conditional means of the responses given the random effects are specified a first step estimator 0 of the vector 0 of parameters is obtained by solving a system of estimating equations it is shown that 9 is consistent if furthermore the conditional variances are correctly specified a second step estimator 0 can be obtained by solving a system of optimal estimating equations whose coefficients a are estimated by 0 it is shown that 0 maintains the asymptotic optimality simulations a also indicate solid improvement of 0 over 0 two examples involving real data analysis are considered the methods are developed under a more general framework so that they may be applied to more general estimation problems

non parametric small area estimation using penalized spline regression the paper proposes a small area estimation approach that combines small area random effects with a smooth non parametrically specified trend by using penalized splines as the representation for the non parametric trend it is possible to express the non parametric small area estimation problem as a mixed effect model regression the resulting model is readily fitted by using existing model fitting approaches such as restricted maximum likelihood we present theoretical results on the prediction mean squared error of the estimator proposed and on likelihood ratio tests for random effects and we propose a simple non parametric bootstrap approach for model inference and estimation of the small area prediction mean squared error the applicability of the method is demonstrated on a survey of lakes in north eastern usa

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

topological sectors and measures on moduli space in quantum yang mills on a riemann surface previous path integral treatments of yang mills on a riemann surface automatically sum over principal fiber bundles of all possible topological types in computing quantum expectations this paper extends the path integral formulation to treat separately each topological sector the formulation is sufficiently explicit to calculate wilson line expectations exactly further it suggests two new measures on the moduli space of flat connections one of which proves to agree with the small volume limit of the yang mills measure

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

generalized linear models for small area estimation abstract bayesian methods have been used quite extensively in recent years for solving small area estimation problems particularly effective in this regard has been the hierarchical or empirical bayes approach which is especially suitable for a systematic connection of local areas through models however the development to date has mainly concentrated on continuous valued variates often the survey data are discrete or categorical so that hierarchical or empirical bayes techniques designed for continuous variates are inappropriate this article considers hierarchical bayes generalized linear models for a unified analysis of both discrete and continuous data a general theorem is provided that ensures the propriety of posteriors under diffuse priors this result is then extended to the case of spatial generalized linear models the hierarchical bayes procedure is implemented via markov chain monte carlo integration techniques two examples one featuring spatial correlation structure are given to illu

multiscale detection of localized anomalous structure in aggregate disease incidence data we present a modelling framework for detection of potentially anomalous structure in aggregate spatial disease incidence data in a manner sensitive to localization at multiple scales and or positions the key technical contribution is the re casting of the components of a multiscale disease mapping methodology recently introduced by the authors in an earlier paper into a form appropriate for hypothesis testing in particular we describe how hypotheses of spatially clustered variations in disease incidence may be linked in one to one correspondence with collections of hypotheses on the values of certain multiscale parameters associated with a user defined hierarchy of nested partitions of an overall spatial region a bayesian hypothesis testing methodology is developed in the context of a standard poisson measurement model over the collection of possible multiscale hypotheses we discuss the specification of hyper parameters and prior distributions on the space of models the methodology is illustrated on both simulated and real data

bayesian prediction in linear models applications to small area estimation this paper introduces a hierarchical bayes hb approach for prediction in general mixed linear models the results find application in small area estimation our model unifies and extends a number of models previously considered in this area computational formulas for obtaining the bayes predictors and their standard errors are given in the general case the methods are applied to two actual data sets also in a special case the hb predictors are shown to possess some interesting frequentist properties

framed sheaves on projective stacks abstract given a normal projective irreducible stack x over an algebraically closed field of characteristic zero we consider framed sheaves on x i e pairs e e where e is a coherent sheaf on x and e is a morphism from e to a fixed coherent sheaf f after introducing a suitable notion of semi stability we construct a projective scheme which is a moduli space for semistable framed sheaves with fixed hilbert polynomial and an open subset of it which is a fine moduli space for stable framed sheaves if x is a projective irreducible orbifold of dimension two and f a locally free sheaf on a smooth divisor d x satisfying certain conditions we consider d f framed sheaves i e framed sheaves e e with e a torsion free sheaf which is locally free in a neighbourhood of d and e d an isomorphism these pairs are stable for a suitable choice of a parameter entering the semi stability condition and of the polarization of x this implies the existence of a fine moduli space parameterizing isomorphism classes of d f framed sheaves on x with fixed hilbert polynomial which is a quasi projective scheme in an appendix we develop the example of stacky hirzebruch surfaces this is the first paper of a project aimed to provide an algebro geometric approach to the study of gauge theories on a wide class of 4 dimensional riemannian manifolds by means of framed sheaves on stacky compactifications of them in particular in a subsequent paper 20 these results are used to study gauge theories on ale spaces of type a k

bayesian estimates of disease maps how important are priors in the fully bayesian fb approach to disease mapping the choice of the hyperprior distribution of the dispersion parameter is a key issue in this context we investigated the sensitivity of the rate ratio estimates to the choice of the hyperprior via a simulation study we also compared the performance of the fb approach to mapping disease risk to the conventional approach of mapping maximum likelihood ml estimates and p values the study was modelled on the incidence data of insulin dependent diabetes mellitus iddm as observed in the communes of sardinia