on measuring sensitivity to parametric model misspecification summary in settings where parametric inference is inconsistent under model misspecification the discrepancy between correct and misspecified inferences is compared with the discrepancy between correct and misspecified models to make the comparison tractable large sample and small misspecification approximations are employed the ratio of the approximate discrepancy between inferences to the approximate discrepancy between models is regarded as a relative measure of sensitivity to model misspecification the maximum ratio over a family of correct distributions is determined as a measure of worst case sensitivity as well the distribution producing this maximum can be examined to see how a particular combination of a parametric family and estimand is susceptible to model misspecifications statisticians recognize that in reality no models are really correct in many parametric situations model misspecifications give rise to asymptotically inconsistent estimators of parameters the goal of this work is to quantify how bad an estimator can be relative to the degree of model misspecification the rationale for such an investigation is that misleading inferences arise when a bad answer is produced from an only mildly misspecified model we are less concerned about bad inferences arising from grossly misspecified models as such models are likely to be screened out by diagnostic checks we quantify the sensitivity of a parametric family and estimand to model misspecifications by comparing the kullback leibler kl divergence between correct and misspecified inferences with the kl divergence between correct and misspecified models more specifically the divergence between inferences can be based on either sampling distributions of maximum likelihood estimators or posterior distributions for tractability local approx imations to the kl divergence are employed in a somewhat similar spirit to cook 1986 the maximum ratio of approximate divergences over a class of correct models can be com puted as a worst case measure of sensitivity to misspecifications as well the correct model achieving the maximum can be examined to gain insight into the kind of departures from the model which are damaging this methodology is described fully in the next section then section 3 details an application to some common parametric models for survival data the paper concludes with a discussion in section 4 some technical comments are relegated to appendix a and appendix b

semiparametric bayesian analysis of selection models selection models are appropriate when a datum x enters the sample only with probability or weight w x it is typically assumed that the weight function w is monotone but the precise functional form of the weight function is often unknown in this article the dirichlet process prior centered on a parametric form is used as a prior distribution on the weight function this allows for incorporation of knowledge about the weight function without restricting it to be of some particular functional form by introducing latent variables related to the selection mechanism computation via gibbs sampling can be implemented in the case where the total number of selected and unselected observations n is known when n is unknown a reversible jump markov chain sampler is needed to carry out the computations an important difficulty that can be thought of as practical nonidentifiability is revealed even for selection models in which the weight functions are theoretically identifiable the proposed solution t

an effective method for solving the hyper singular integral equations in 3 d acoustics the application of the boundary integral methods to the problem of acoustics exterior to a three dimensional surface suffers in general from the nonexistence of nonuniqueness of its solutions at frequencies that are characteristic of the associated interior problem the formulation that is most suitable for numerical implementation still appears to be that proposed by burton and miller proc r soc london ser a 323 201 210 1971 however the hypersingular kernels present in such a formulation render it computationally unattractive previous attempts to regularize such hypersingular kernels involved the use of double surface integrals or implicit use of tangential operators or closed form evaluations of hypersingular integrals the method of double surface integrals is computationally highly inefficient even though it allows higher order interpolation schemes on the surface the other two approaches are more conducive to the assumption of a constant value for each of the variables pressure a

on inconsistent bayes estimates of location on montre que dans certaines situations relativement naturelles les estimateurs de position bayesiens sont inconsistants

a consistent nonparametric test of parametric regression functional form in fixed effects panel data models abstract we propose a consistent test for a linear functional form against a nonparametric alternative in a fixed effects panel data model we show that the test has a limiting standard normal distribution under the null hypothesis and show that the test is a consistent test we also establish the asymptotic validity of a bootstrap procedure which is used to better approximate the finite sample null distribution of the test statistic simulation results show that the proposed test performs well for panel data with a large number of cross sectional units and a finite number of observations across time

the incidental parameter problem since 1948 this paper was written to mark the 50th anniversary of neyman and scott s econometrica paper defining the incidental parameter problem it surveys the history both of the paper and of the problem in the statistics and econometrics literature

the use of chief to obtain unique solutions for acoustic radiation using boundary integral equations this report is concerned with the problem of obtaining a unique solution for radiation problems at characteristic wavenumbers when a boundary integral equation formulation is used it is shown that the combined helmholtz integral equation formulation chief method works even when numerous interior points lie exactly on nodal surfaces of a mode corresponding to an eigenfrequency of the related interior problem when at least one interior point does not lie on a nodal surface the use of chief is illustrated with numerical radiation experiments for the sphere and the finite cylinder chief is compared to the helmholtz gradient formulation hgf for circumventing nonuniqueness and is found to yield a more accurate solution in at least one published example a procedure is used to indicate the solution error when using integral equation methods with or without a technique to circumvent nonuniqueness this procedure uses the velocity potential of an interior point as an indicator of solution error in all cases considered the interior potential correctly indicated a good or bad solution whereas the matrix condition number falsely indicated a bad solution in several instances work supported in part by onr

bayes and empirical bayes do they merge bayesian inference is attractive for its coherence and good frequentist properties however eliciting a honest prior may be difficult and a common practice is to take an empirical bayes approach using some empirical estimate of the prior hyperparameters despite not rigorous the underlying idea is that for sufficiently large sample size empirical bayes leads to similar inferential answers as a proper bayesian inference however precise mathematical results seem missing in this work we give more rigorous results in terms of merging of bayesian and empirical bayesian posterior distributions we study two notions of merging bayesian weak merging and frequentist merging in total variation we also show that under regularity conditions empirical bayes asymptotically gives an oracle selection of the prior hyperparameters examples include empirical bayes density estimation with dirichlet process mixtures

on the consistency of bayes estimates on etudie les proprietes de frequence des regles de bayes avec une attention particuliere a la consistence

bayesian methods and maximum entropy for ill posed inverse problems in this paper we study linear inverse problems where some generalized moments of an unknown positive measure are observed we introduce a new construction called the maximum entropy on the mean method mem which relies on a suitable sequence of finite dimensional discretized inverse problems its advantage is threefold it allows us to interpret all usual deterministic methods as bayesian methods it gives a very convenient way of taking into account prior information it also leads to new criteria for the existence question concerning the linear inverse problem which will be a starting point for the investigation of superresolution phenomena the key tool in this work is the large deviations property of some discrete random measure connected with the reconstruction procedure

on bayesian consistency we consider a sequence of posterior distributions based on a data dependent prior which we shall refer to as a pseudoposterior distribution and establish simple conditions under which the sequence is hellinger consistent it is shown how investigations into these pseudo posteriors assist with the understanding of some true posterior distributions including polya trees the infinite dimensional exponential family and mixture models

bayesian goodness of fit testing using infinite dimensional exponential families we develop a nonparametric bayes factor for testing the fit of a parametric model we begin with a nominal parametric family which we then embed into an infinite dimensional exponential family the new model then has a parametric and nonparametric component we give the log density of the nonparametric component a gaussian process prior an asymptotic consistency requirement puts a restriction on the form of the prior leaving us with a single hyperparameter for which we suggest a default value based on simulation experience then we construct a bayes factor to test the nominal model versus the semiparametric alternative finally we show that the bayes factor is consistent the proof of the consistency is based on approximating the model by a sequence of exponential families

information consistency of nonparametric gaussian process methods bayesian nonparametric models are widely and successfully used for statistical prediction while posterior consistency properties are well studied in quite general settings results have been proved using abstract concepts such as metric entropy and they come with subtle conditions which are hard to validate and not intuitive when applied to concrete models furthermore convergence rates are difficult to obtain by focussing on the concept of information consistency for bayesian gaussian process gp models consistency results and convergence rates are obtained via a regret bound on cumulative log loss these results depend strongly on the covariance function of the prior process thereby giving a novel interpretation to penalization with reproducing kernel hilbert space norms and to commonly used covariance function classes and their parameters the proof of the main result employs elementary convexity arguments only a theorem of widom is used in order to obtain precise convergence rates for several covariance functions widely used in practice

the semiparametric bernstein von mises theorem in a smooth semiparametric estimation problem the marginal posterior for the parameter of interest is expected to be asymptotically normal and satisfy frequentist criteria of optimality if the model is endowed with a suitable prior it is shown that under certain straightforward and interpretable conditions the assertion of le cam s acclaimed but strictly parametric bernstein von mises theorem univ california publ statist 1 1953 277 329 holds in the semiparametric situation as well as a consequence bayesian point estimators achieve efficiency for example in the sense of hajek s convolution theorem z wahrsch verw gebiete 14 1970 323 330 the model is required to satisfy differentiability and metric entropy conditions while the nuisance prior must assign nonzero mass to certain kullback leibler neighborhoods ghosal ghosh and van der vaart ann statist 28 2000 500 531 in addition the marginal posterior is required to converge at parametric rate which appears to be the most stringent condition in examples the results are applied to estimation of the linear coefficient in partial linear regression with a gaussian prior on a smoothness class for the nuisance

dirichlet process mixtures of generalized linear models we propose dirichlet process mixtures of generalized linear models dp glm a new class of methods for nonparametric regression given a data set of input response pairs the dp glm produces a global model of the joint distribution through a mixture of local generalized linear models dp glms allow both continuous and categorical inputs and can model the same class of responses that can be modeled with a generalized linear model we study the properties of the dp glm and show why it provides better predictions and density estimates than existing dirichlet process mixture regression models we give conditions for weak consistency of the joint distribution and pointwise consistency of the regression estimate

bayesian inference for semiparametric regression using a fourier representation summary this paper presents the bayesian analysis of a semiparametric regression model that consists of parametric and nonparametric components the nonparametric component is represented with a fourier series where the fourier coefficients are assumed a priori to have zero means and to decay to 0 in probability at either algebraic or geometric rates the rate of decay controls the smoothness of the response function the posterior analysis automatically selects the amount of smoothing that is coherent with the model and data posterior probabilities of the parametric and semiparametric models provide a method for testing the parametric model against a non specific alternative the bayes estimator s mean integrated squared error compares favourably with the theoretically optimal estimator for kernel regression

consistent and robust bayes procedures for location based on partial information we consider bayes procedures for a location parameter that are robust with respect to the shape of the distribution f of the data the case where f is fixed and the case where f has a dirichlet distribution are both treated the procedures are based on the posterior distributions of the location parameter given the partial information contained in a robust estimate of location we show consistency and asymptotic normality of the procedures and give instances where the bayes procedure based on the full sample diverges while the bayes procedures based on partial information converges and is asymptotically normal

a weighted residual formulation for the chief method in acoustics in this article the combined helmholtz integral equation formulation chief proposed by schenck is greatly improved so that it will alleviate the annoying nodal surface problem the constraint provided by each chief equation and its first order derivatives is enforced in a weighted residual sense over a small interior region rather than only at a single point consequently chief blocks rather than chief points are used in the numerical implementation although the corresponding interior problem may have nodal surfaces at characteristic frequencies it has no such things as nodal blocks therefore the chief block chief b method is expected to work even if all of the chief blocks fall on the intersection of several nodal surfaces this new formulation is implemented in an isoparametric element environment numerical examples in spherical radiation and scattering at intermediate frequencies are given to test the new formulation

asymptotics for posterior hazards an important issue in survival analysis is the investigation and the modeling of hazard rates within a bayesian nonparametric framework a natural and popular approach is to model hazard rates as kernel mixtures with respect to a completely random measure in this paper we provide a comprehensive analysis of the asymptotic behavior of such models we investigate consistency of the posterior distribution and derive fixed sample size central limit theorems for both linear and quadratic functionals of the posterior hazard rate the general results are then specialized to various specific kernels and mixing measures yielding consistency under minimal conditions and neat central limit theorems for the distribution of functionals

data tracking and the understanding of bayesian consistency summary we deal with strong consistency for bayesian density estimation an awkward con sequence of inconsistency is described it is pointed out that consistency at some density fo depends on the prior mass assigned to the pathological set of those densities that are close to fo in a weak sense and far apart from fo in a hellinger sense an analysis of these sets leads to the identification of the notion of data tracking specific examples in which this phenomenon cannot occur are discussed when it can happen we show how and where things can go wrong thus providing more intuition about the sources of inconsistency

classical subjective expected utility we consider decision makers who know that payoff relevant observations are generated by a process that belongs to a given class m as postulated in wald wald a 1950 statistical decision functions wiley new york we incorporate this waldean piece of objective information within an otherwise subjective setting a la savage savage lj 1954 the foundations of statistics wiley new york and show that this leads to a two stage subjective expected utility model that accounts for both state and model uncertainty

robust bayesian inference in elliptical regression models broadening the stochastic assumptions on the error terms of regression models was prompted by the analysis of linear multivariate t models in zellner 1976 we consider a possibly nonlinear regression model under any multivariate elliptical data density and examine bayesian posterior and predictive results the latter are shown to be robust with respect to the specific choice of a sampling density within this elliptical class in particular sufficient conditions for such model robustness are that we single out a precision factor 2 on which we can specify an improper prior density apart from theb posterior distribution of this nuisance parameter 2 the entire analysis will then be completely unaffected by departures from normality similar results hold in finite mixtures of such elliptical densities which can be used to average out specification uncertainty

a numerical study of the coefficient matrix of the boundary element method near characteristic frequencies abstract in this paper the well known failure in numerical computations of the exterior surface helmholtz integral equation at certain characteristic frequencies is investigated the problem of characteristic frequencies is numerically reflected in an ill conditioning of the coefficient matrix arising from the transformation of the integral equation to a linear set of equations by means of a rank revealing factorization the rank deficiency of the coefficient matrix is determined and thereby the necessary number of equations to be added to the original system of equations can be found one of the most popular methods to overcome this non uniqueness problem is the combined helmholtz integral equation formulation chief proposed by schenck 1 however chief points placed on or near a nodal surface of the corresponding interior problem do not provide a linearly independent constraint and the problem of selecting good chief points is still a topic of considerable interest in this paper the chief is used with a rank revealing factorization and it is shown that the necessary number of good chief points can be predicted furthermore a method of determining whether a chief point is good is presented the new approach has been tested on an axisymmetric bem formulation with one or several chief points

thermodynamics of fiber bundles a recent theory that determines the properties of disordered solids as the solid accumulates damage is applied to the special case of fiber bundles with global load sharing and is shown to be exact in this case the theory postulates that the probability of observing a given emergent damage state is obtained by maximizing the emergent entropy as defined by shannon subject to energetic constraints this theory yields the known exact results for the fiber bundle model with global load sharing and holds for any quenched disorder distribution it further defines how the entropy evolves as a function of stress and shows definitively how the concepts of temperature and entropy emerge in a problem where all statistics derive from the initial quenched disorder a previously unnoticed phase transition is shown to exist as the entropy goes through a maximum in general this entropy maximum transition occurs at a different point in strain history than the stress maximum transition with the precise location depending entirely on the quenched disorder distribution

finite sample simulation based inference in var models with application to granger causality testing tests in vector autoregressive var models are typically based on large sample approximations involving the use of asymptotic distributions or bootstrap techniques after documenting that such methods can be very misleading even with fairly large samples we propose a general simulation based technique that allows one to control completely test levels in parametric var models in particular we show that maximized monte carlo tests dufour 2005 monte carlo tests with nuisance parameters a general approach to finite sample inference and nonstandard asymptotics in econometrics journal of econometrics forthcoming can provide provably exact tests for such models whether they are stationary or integrated applications to order selection and causality testing are considered as special cases the technique developed is applied to a var model of the u s economy

bayesian nonparametric estimation of the median part i computation of the estimates let x i i 1 ldots n be i i d sim f theta where f theta x f x theta for some f that has median equal to 0 f is assumed unknown or only partially known and the problem is to estimate theta priors are put on the pair f theta the priors on f concentrate all their mass on c d f s with median equal to 0 these priors include dirichlet type priors the marginal posterior distribution of theta given x 1 ldots x n is computed the mean of the posterior is taken as the estimate of theta

conditioning likelihood and coherence a review of some foundational concepts 2000 conditioning likelihood and coherence a review of some foundational concepts journal of the american statistical association vol 95 no 452 pp 1340 1346

adaptive estimation of the threshold point in threshold regression this paper studies semiparametric efficient estimation of the threshold point in threshold regression the classical literature of semiparametric efficient estimation rests on the fact that the maximum likelihood estimator is efficient in any parametric submodel for a large class of loss functions however in threshold regression the maximum likelihood estimator is not efficient while the bayes estimators are efficient and different loss functions induce different efficient estimators for an additively separable loss function that separates the efficiency problem of the threshold point from that of other parameters we show that the semiparametric and parametric efficiency risk bounds coincide then we design a semiparametric empirical bayes estimator to achieve this bound in consequence the threshold point can be adaptively estimated even under conditional moment restrictions we also provide a valid confidence interval called the nonparametric posterior interval for the threshold point simulation studies show that the semiparametric empirical bayes approach is substantially better than existing methods to illustrate our procedure in practice we apply it to an economic growth model for detecting different growth patterns

statistical aspects of the continuous damage theory abstract the present study addresses several fundamental aspects of a link connecting the continuous damage theory and the statistical strength theories the attention is focused on a simple but efficient way of devising rational damage laws based on the physical nature of the phenomenon the computed results evidence the remarkable flexibility of the suggested model in its application for a wide range of different materials

a method to overcome computational difficulties in the exterior acoustics problem when the boundary integral equation method is applied to exterior acoustics problems singularities occur in the resulting algebraic equations at various frequencies associated with the eigenvalues of an interior problem these frequencies are referred to as forbidden and various methods have been devised to overcome the computational difficulties presented at these frequencies the work presented here is an extension to the chief method in that higher derivatives in addition to the function itself are constrained to be zero at selected points in the interior domain whereas the relative success of either method depends on the quantity and selection of interior points the superchief method requires fewer interior points and is less sensitive to point selection resulting in improved robustness without a significant increase in computational complexity

merging and testing opinions we study the merging and the testing of opinions in the context of a prediction model in the absence of incentive problems opinions can be tested and rejected regardless of whether or not data produces consensus among bayesian agents in contrast in the presence of incentive problems opinions can only be tested and rejected when data produces consensus among bayesian agents these results show a strong connection between the testing and the merging of opinions they also relate the literature on bayesian learning and the literature on testing strategic experts 1 introduction data can produce consensus among bayesian agents who initially disagree it can also test and reject opinions we relate these two critical uses of data in a model where agents may strategically misrepresent what they know in each period either 0 or 1 is observed let p and q be two probability measures on 0 1 1 such that q is absolutely continuous with respect to p if p and q are additive then as shown by blackwell and dubins 1962 the conditional probabilities of p and q merge in the sense that the two posteriors become uniformly close as the amount of observations increases q almost surely so repeated applications of bayes rule lead to consensus among bayesian agents provided that their opinions were initially compatible now consider savage s axiomatization of subjective probability he proposed postulates that characterize a preference relation over bets in terms of a nonatomic finitely additive probability p call such p for short an opinion savage s framework allows for finitely additive probabilities that are not additive in particular the conclusions of the blackwell and dubins theorem hold for some but not all opinions this flexibility makes savage s

on finding predictors for arbitrary families of processes the problem is sequence prediction in the following setting a sequence x1 xn of discrete valued observations is generated according to some unknown probabilistic law measure after observing each outcome it is required to give the conditional probabilities of the next observation the measure belongs to an arbitrary but known class c of stochastic process measures we are interested in predictors whose conditional probabilities converge in some sense to the true conditional probabilities if any c is chosen to generate the sequence the contribution of this work is in characterizing the families c for which such predictors exist and in providing a specific and simple form in which to look for a solution we show that if any predictor works then there exists a bayesian predictor whose prior is discrete and which works too we also find several sufficient and necessary conditions for the existence of a predictor in terms of topological characterizations of the family c as well as in terms of local behaviour of the measures in c which in some cases lead to procedures for constructing such predictors it should be emphasized that the framework is completely general the stochastic processes considered are not required to be i i d stationary or to belong to any parametric or countable family

a semiparametric bernstein von mises theorem for gaussian process priors this paper is a contribution to the bayesian theory of semiparametric estimation we are interested in the so called bernstein von mises theorem in a semiparametric framework where the unknown quantity is f with the parameter of interest and f an infinite dimensional nuisance parameter two theorems are established one in the case with no loss of information and one in the information loss case with gaussian process priors the general theory is applied to three specific models the estimation of the center of symmetry of a symmetric function in gaussian white noise a time discrete functional data analysis model and cox s proportional hazards model in all cases the range of application of the theorems is investigated by using a family of gaussian priors parametrized by a continuous parameter

bayesian nonparametric estimation for incomplete data via successive substitution sampling in the problem of estimating an unknown distribution function f in the presence of censoring one can use a nonparametric estimator such as the kaplan meier estimator or one can consider parametric modeling there are many situations where physical reasons indicate that a certain parametric model holds approximately in these cases a nonparametric estimator may be very inefficient relative to a parametric estimator on the other hand if the parametric model is only a crude approximation to the actual model then the parametric estimator may perform poorly relative to the nonparametric estimator and may even be inconsistent the bayesian paradigm provides a reasonable framework for this problem in a bayesian approach one would try to put a prior distribution on f that gives most of its mass to small neighborhoods of the entire parametric family we show that certain priors based on the dirichlet process prior can be used to accomplish this for these priors the posterior distribution of f given the censored data appears to be analytically intractable we provide a method for approximating this posterior distribution through the use of a successive substitution sampling algorithm we also show convergence of the algorithm

exact sample conditioned mse performance of the bayesian mmse estimator for classification error part ii consistency and performance analysis in part i of a two part study on the mse performance of bayesian error estimation we have derived analytical expressions for mse conditioned on the sample for bayesian error estimators and arbitrary error estimators in two bayesian models discrete classification with dirichlet priors and linear classification of gaussian distributions with normal inverse wishart priors here in part ii we examine the consistency of bayesian error estimation and provide several simulation studies that illustrate the concept of conditional mse and how it may be used in practice a salient application is censored sampling where sample points are collected one at a time until the conditional mse reaches a stopping criterion

isogeometric indirect boundary element method for solving the 3d acoustic problems abstract this paper provides an isogeometric indirect boundary element method igibem based on nurbs non uniform rational b splines and pht splines polynomial splines over hierarchical t meshes for analyzing the three dimensional 3d acoustic problems in the classical procedure the geometries are discretized by lagrange polynomials elements which leads to both substantial geometrical error and time consuming meshing steps however these deficiencies can be eliminated by the isogeometric analysis iga directly incorporating the geometry description generated from the cad computer aided design software into cae computer aided engineering analysis unlike the dbem direct boundary element method the ibem indirect boundary element method allows different types of boundary conditions on the two sides of a surface moreover the hypersingular integrals in ibem can be transformed to a weakly singular form in addition the pht based ibem is used to investigate the influence of local refinement on the accuracy of solutions finally the non uniqueness problem is solved which is a fatal defect in the acoustic bem for the exterior problem four different methods to handle the non uniqueness problem are discussed and compared the results obtained by the proposed method were compared with analytical solutions and the results computed by lagrange based ibem several benchmark examples demonstrate 1 the present method i e igibem has super accuracy over conventional ibem for the acoustic problems 2 local refinement has a significant influence on the convergence rate of the solutions and the numerical accuracy is relevant to the distance to the boundary where the local refinement acted 3 as for the non uniqueness problem the imposition of specific interior boundary conditions not only obtains the best calculation result over the entire range of frequencies but also has a simple integral formulation

a boundary element method for acoustic radiation valid for all wavenumbers a boundary element method for solving the exterior acoustic radiation problem that is valid for all wavenumbers has been developed the new formulation derives from the earlier work of burton and miller proc r soc london ser a 323 201 210 1971 which uses a linear combination of the helmholtz integral equation and its gradient to overcome the uniqueness problem a main feature of this formulation is the absence of integral singularities by restricting the field points of the kernels of the integrals to interior regions well away from radiating boundaries the usual complication of dealing with singular kernels is circumvented the new formulation has been named chi for coupled helmholtz integrals a corresponding computer program chi 4 35 can be used to compute the surface acoustic pressure on an arbitrary body for a given specified surface normal velocity results generated with chi 4 35 are presented for the spherical monopole and dipole along with results for parametric studies on the acc

consistency of bernstein polynomial posteriors a bernstein prior is a probability measure on the space of all the distribution functions on 0 1 under very general assumptions it selects absolutely continuous distribution functions whose densities are mixtures of known beta densities the bemstein prior is of interest in bayesian nonparametric inference with continuous data we study the consistency of the posterior from a bernstein prior we first show that under mild assumptions the posterior is weakly consistent for any distribution function p o on 0 1 with continuous and bounded lebesgue density with slightly stronger assumptions on the prior the posterior is also hellinger consistent this implies that the predictive density from a bernstein prior which is a bayesian density estimate converges in the hellinger sense to the true density assuming that it is continuous and bounded we also study a sieve maximum likelihood version of the density estimator and show that it is also hellinger consistent under weak assumptions when the order of the bernstein polynomial i e the number of components in the beta distribution mixture is truncated we show that under mild restrictions the posterior concentrates on the set of pseudotrue densities finally we study the behaviour of the predictive density numerically and we also study a hybrid bayes maximum likelihood density estimator

local sensitivity of posterior expectations we investigate the degree to which posterior expectations are sensitive to prior distributions using a local method based on functional differentiation invariance considerations suggest a family of norms which can be used to measure perturbations to the prior the sensitivity measure is seen to depend heavily on the choice of norm asymptotic results suggest which norm will yield the most useful results in practice we find that to maintain asymptotically sensible behaviour it is necessary to reduce the richness of the class of prior perturbations as the dimension of the parameter space increases jeffreys prior is characterized as the prior to which inference is least sensitive

filtered variate prior distributions for histogram smoothing abstract we develop prior distributions for histogram inference favoring smooth population frequencies that is probability vectors with small differences for neighboring categories we give a theory of prior random probability vectors representable as a linear transform or filter of a standard random probability vector or equivalently a random weighted average of nonrandom smooth probability vectors promising methods of prior assessment are given based on elicitation of a list of typically smooth probability vectors the empirical moments of which can then be matched by the mean vector and variance matrix of a constructed continuous type filtered variate prior distribution

adaptive bernstein von mises theorems in gaussian white noise we investigate bernstein von mises theorems for adaptive nonparametric bayesian procedures in the canonical gaussian white noise model we consider both a hilbert space and multiscale setting with applications in l 2 and l infty respectively this provides a theoretical justification for plug in procedures for example the use of certain credible sets for sufficiently smooth linear functionals we use this general approach to construct optimal frequentist confidence sets based on the posterior distribution we also provide simulations to numerically illustrate our approach and obtain a visual representation of the geometries involved

a bayesian goodness of fit test and semiparametric generalization of logistic regression with measurement data summary logistic regression is a popular tool for risk analysis in medical and population health science with continuous response data it is common to create a dichotomous outcome for logistic regression analysis by specifying a threshold for positivity fitting a linear regression to the nondichotomized response variable assuming a logistic sampling model for the data has been empirically shown to yield more efficient estimates of odds ratios than ordinary logistic regression of the dichotomized endpoint we illustrate that risk inference is not robust to departures from the parametric logistic distribution moreover the model assumption of proportional odds is generally not satisfied when the condition of a logistic distribution for the data is violated leading to biased inference from a parametric logistic analysis we develop novel bayesian semiparametric methodology for testing goodness of fit of parametric logistic regression with continuous measurement data the testing procedures hold for any cutoff threshold and our approach simultaneously provides the ability to perform semiparametric risk estimation bayes factors are calculated using the savage dickey ratio for testing the null hypothesis of logistic regression versus a semiparametric generalization we propose a fully bayesian and a computationally efficient empirical bayesian approach to testing and we present methods for semiparametric estimation of risks relative risks and odds ratios when parametric logistic regression fails theoretical results establish the consistency of the empirical bayes test results from simulated data show that the proposed approach provides accurate inference irrespective of whether parametric assumptions hold or not evaluation of risk factors for obesity shows that different inferences are derived from an analysis of a real data set when deviations from a logistic distribution are permissible in a flexible semiparametric framework

bayesian inference for semiparametric binary regression abstract we propose a regression model for binary response data that places no structural restrictions on the link function except monotonicity and known location and scale predictors enter linearly we demonstrate bayesian inference calculations in this model by modifying the dirichlet process we obtain a natural prior measure over this semiparametric model and we use polya sequence theory to formulate this measure in terms of a finite number of unobserved variables we design a markov chain monte carlo algorithm for posterior simulation and apply the methodology to data on radiotherapy treatments for cancer

are gibbs type priors the most natural generalization of the dirichlet process discrete random probability measures and the exchangeable random partitions they induce are key tools for addressing a variety of estimation and prediction problems in bayesian inference here we focus on the family of gibbs type priors a recent elegant generalization of the dirichlet and the pitman yor process priors these random probability measures share properties that are appealing both from a theoretical and an applied point of view i they admit an intuitive predictive characterization justifying their use in terms of a precise assumption on the learning mechanism ii they stand out in terms of mathematical tractability iii they include several interesting special cases besides the dirichlet and the pitman yor processes the goal of our paper is to provide a systematic and unified treatment of gibbs type priors and highlight their implications for bayesian nonparametric inference we deal with their distributional properties the resulting estimators frequentist asymptotic validation and the construction of time dependent versions applications mainly concerning mixture models and species sampling serve to convey the main ideas the intuition inherent to this class of priors and the neat results they lead to make one wonder whether it actually represents the most natural generalization of the dirichlet process

asymptotics for statistical treatment rules this paper develops asymptotic optimality theory for statistical treatment rules in smooth parametric and semiparametric models manski 2000 2002 2004 and dehejia 2005 have argued that the problem of choosing treatments to maximize social welfare is distinct from the point estimation and hypothesis testing problems usually considered in the treatment eects literature and advocate formal analysis of decision procedures that map empirical data into treatment choices we develop large sample approximations to statistical treatment assignment problems in both randomized experiments and observational data settings in which treatment eects are identified we derive a local asymptotic minmax regret bound on social welfare and a local asymptotic risk bound for a two point loss function we show that certain natural treatment assignment rules attain these bounds

a simple method avoiding non uniqueness in the boundary element method for acoustic scattering problem the boundary element method bem is widely used for sound field analysis problems however it has a non uniqueness problem in the exterior domain various methods to avoid this problem have been developed however these are not easily applied to the bem in this paper a simple method called the ica ring inner cavity ringing method is proposed for avoiding the non uniqueness problem and this method is applied to the bem in both single and plural domains the concept of the ica ring method is that a scatterer in free space is hollowed as a shell and the volume is smaller the eigenfrequencies are shifted to a higher range next the mechanism of the non uniqueness problem in plural domains and a reason of the application of the ica ring method to the case of plural domains are explained finally some results calculated by the bem using the ica ring method are shown the calculational condition is that a cylinder with radius 0 125m floats in two dimensional free space in this case no calculationa

on the convergence of the laplace approximation and noise level robustness of laplace based monte carlo methods for bayesian inverse problems the bayesian approach to inverse problems provides a rigorous framework for the incorporation and quantification of uncertainties in measurements parameters and models we are interested in designing numerical methods which are robust w r t the size of the observational noise i e methods which behave well in case of concentrated posterior measures the concentration of the posterior is a highly desirable situation in practice since it relates to informative or large data however it can pose a computational challenge for numerical methods based on the prior measure we propose to employ the laplace approximation of the posterior as the base measure for numerical integration in this context the laplace approximation is a gaussian measure centered at the maximum a posteriori estimate and with covariance matrix depending on the logposterior density we discuss convergence results of the laplace approximation in terms of the hellinger distance and analyze the efficiency of monte carlo methods based on it in particular we show that laplace based importance sampling and laplace based quasi monte carlo methods are robust w r t the concentration of the posterior for large classes of posterior distributions and integrands whereas prior based importance sampling and plain quasi monte carlo are not numerical experiments are presented to illustrate the theoretical findings

solving the hypersingular boundary integral equation in three dimensional acoustics using a regularization relationship regularization of the hypersingular integral in the normal derivative of the conventional helmholtz integral equation through a double surface integral method or regularization relationship has been studied by introducing the new concept of discretized operator matrix evaluation of the double surface integrals is reduced to calculate the product of two discretized operator matrices such a treatment greatly improves the computational efficiency as the number of frequencies to be computed increases the computational cost of solving the composite helmholtz integral equation is comparable to that of solving the conventional helmholtz integral equation in this paper the detailed formulation of the proposed regularization method is presented the computational efficiency and accuracy of the regularization method are demonstrated for a general class of acoustic radiation and scattering problems the radiation of a pulsating sphere an oscillating sphere and a rigid sphere insonified by a plane acoustic wave are solved using the new method with curvilinear quadrilateral isoparametric elements it is found that the numerical results rapidly converge to the corresponding analytical solutions as finer meshes are applied

bayesian nonparametric inference for random distributions and related functions in recent years bayesian nonparametric inference both theoretical and computational has witnessed considerable advances however these advances have not received a full critical and comparative analysis of their scope impact and limitations in statistical modelling many aspects of the theory and methods remain a mystery to practitioners and many open guestions remain in this paper we discuss and illustrate the rich modelling and analytic possibilities that are available to the statistician within the bayesian nonparametric and or semiparametric framework

a boundary integral equation method for two dimensional acoustic scattering problems this work formulates the singularity free integral equations to study 2 d acoustic scattering problems to avert the nonuniqueness difficulties burton s and burton and miller s methods are employed to solve the dirichlet and neumann problems respectively the surface helmholtz integral equations and their normal derivative equations in bounded form are derived the weakly singular integrals are desingularized by subtracting a term from the integrand and adding it back with an exact value depending on the relevant problem the additional integral can finally be either expressed in an explicit form or transformed to form a surface source distribution of the related equipotential body the hypersingular kernel is desingularized further using some properties of an interior laplace problem the new formulations are advantageous in that they can be computed by directly using standard quadrature formulas also discussed is the contour a unique feature of 2 d problems instead of dividing the boundary surface into several small elements a parametric representation of a 2 d boundary curve is further proposed to facilitate a global numerical implementation calculations consist of acoustic scattering by a hard and a soft circular cylinder respectively comparing the numerical results with the exact solutions demonstrates the proposed method s effectiveness

catching up faster by switching sooner a predictive approach to adaptive estimation with an application to the aic bic dilemma summary prediction and estimation based on bayesian model selection and model averaging and derived methods such as the bayesian information criterion bic do not always converge at the fastest possible rate we identify the catch up phenomenon as a novel explanation for the slow convergence of bayesian methods which inspires a modification of the bayesian predictive distribution called the switch distribution when used as an adaptive estimator the switch distribution does achieve optimal cumulative risk convergence rates in non parametric density estimation and gaussian regression problems we show that the minimax cumulative risk is obtained under very weak conditions and without knowledge of the underlying degree of smoothness unlike other adaptive model selection procedures such as the akaike information criterion aic and leave one out cross validation bic and bayes factor model selection are typically statistically consistent we show that this property is retained by the switch distribution which thus solves the aic bic dilemma for cumulative risk the switch distribution has an efficient implementation we compare its performance with aic bic and bayesian model selection and averaging on a regression problem with simulated data

frequentist bayesian monte carlo test for mean vectors in high dimension abstract conventional methods for testing the mean vector of a p variate gaussian distribution require a sample size n greater than or equal to p but in high dimensional situations that is when n is smaller than p special and new adjustments are needed although bayesian empirical methods are well succeeded for testing in high dimension their performances are strongly dependent on the actual unknown covariance matrix of the gaussian random vector in this paper we introduce a hybrid frequentist bayesian monte carlo test and prove that i under the null hypothesis the performance of the proposed test is invariant with respect to the real unknown covariance matrix and ii the decision rule is valid which means that in terms of expected loss the performance of the proposed procedure can always be made as good as the exact bayesian test and in terms of type i error probability the method is always of level for arbitrary 0 1

on consistency of nonparametric normal mixtures for bayesian density estimation the past decade has seen a remarkable development in the area of bayesian nonparametric inference from both theoretical and applied perspectives as for the latter the celebrated dirichlet process has been successfully exploited within bayesian mixture models leading to many interesting applications as for the former some new discrete nonparametric priors have been recently proposed in the literature that have natural use as alternatives to the dirichlet process in a bayesian hierarchical model for density estimation when using such models for concrete applications an investigation of their statistical properties is mandatory of these properties a prominent role is to be assigned to consistency indeed strong consistency of bayesian nonparametric procedures for density estimation has been the focus of a considerable amount of research in particular much attention has been devoted to the normal mixture of dirichlet process in this article we improve on previous contributions by establishing str

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

some new results for dirichlet priors let p be a random probability measure chosen by a dirichlet process whose parameter a is a finite measure with support contained in 0 and suppose that v f x 2 p dx f xp dx 2 is a finite random variable this paper deals with the distribution of v which is given in a rather general case a simple application to bayesian bootstrap is also illustrated

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

nonparametric estimation of convex models via mixtures we present a general approach to estimating probability measures constrained to lie in a convex set we represent constrained measures as mixtures of simple known extreme measures and so the problem of estimating a constrained measure becomes one of estimating an unconstrained mixing measure convex constraints arise in many modeling situations such as estimation of the mean and estimation under stochastic ordering constraints we describe mixture representation techniques for these and other situations and discuss applications to maximum likelihood and bayesian estimation

an integral equation approach to three dimensional acoustic radiation and scattering problems this paper provides numerical implementation and tests for a boundary integral equation method proposed recently by krutitskii ima j appl math 64 259 269 2000 for evaluating three dimensional acoustic radiation and scattering fields in the exterior neumann problem for several obstacles auxiliary boundaries with certain prescribed boundary conditions are introduced inside the obstacles for the purpose of eliminating the fictitious characteristic frequencies applying the simple source formulation on all boundaries leads to a set of uniquely solvable integral equations with weakly singular kernels to facilitate the numerical implementation the modified integral equations are further transformed to ordinary ones that contain nonsingular kernels only test cases consist of the acoustic radiation and scattering by one sphere and a pair of equal spheres comparisons are made with analytical solutions and schenck s chief method numerical results demonstrate the efficiency robustness and accuracy of

on inconsistent bayes estimates in the discrete case consider sampling from an unknown probability distribution on the integers with a tail free prior the posterior distribution is consistent with a mixture of a tail free prior and a point mass however the posterior may be inconsistent this is likewise true for a countable mixture of tail free priors similar results are given for dirichlet priors

assessment and propagation of model uncertainty in most examples of inference and prediction the expression of uncertainty about unknown quantities y on the basis of known quantities x is based on a model m that formalizes assumptions about how x and y are related m will typically have two parts structural assumptions s such as the form of the link function and the choice of error distribution in a generalized linear model and parameters whose meaning is specific to a given choice of s it is common in statistical theory and practice to acknowledge parametric uncertainty about given a particular assumed structure s it is less common to acknowledge structural uncertainty about s itself a widely used approach involves enlisting the aid of x to specify a plausible single best choice s for s and then proceeding as if s were known to be correct in general this approach fails to assess and propagate structural uncertainty fully and may lead to miscalibrated uncertainty assessments about y given x when miscalibration occurs it will often result in understatement of inferential or predictive uncertainty about y leading to inaccurate scientific summaries and overconfident decisions that do not incorporate sufficient hedging against uncertainty in this paper i discuss a bayesian approach to solving this problem that has long been available in principle but is only now becoming routinely feasible by virtue of recent computational advances and examine its implementation in examples that involve forecasting the price of oil and estimating the chance of catastrophic failure of the us space shuttle

bernstein von mises theorem for linear functionals of the density in this paper we study the asymptotic posterior distribution of linear functionals of the density by deriving general conditions to obtain a semi parametric version of the bernstein von mises theorem the special case of the cumulative distributive function evaluated at a specific point is widely considered in particular we show that for infinite dimensional exponential families under quite general assumptions the asymptotic posterior distribution of the functional can be either gaussian or a mixture of gaussian distributions with different centering points this illustrates the positive but also the negative phenomena that can occur in the study of bernstein von mises results

mutual information metric entropy and cumulative relative entropy risk assume p theta theta epsilon theta is a set of probability distributions with a common dominating measure on a complete separable metric space y a state theta epsilon theta is chosen by nature a statistician obtains n independent observations y 1 dots y n from y distributed according to p theta for each time t between 1 and n based on the observations y 1 dots y t 1 the statistician produces an estimated distribution hat p t for p theta and suffers a loss l p theta hat p t the cumulative risk for the statistician is the average total loss up to time n of special interest in information theory data compression mathematical finance computational learning theory and statistical mechanics is the special case when the loss l p theta hat p t is the relative entropy between the true distribution p theta and the estimated distribution hat p t here the cumulative bayes risk from time 1 to n is the mutual information between the random parameter theta and the observations y 1 dots y n new bounds on this mutual information are given in terms of the laplace transform of the hellinger distance between pairs of distributions indexed by parameters in theta from these bounds on the cumulative minimax risk are given in terms of the metric entropy of theta with respect to the hellinger distance the assumptions required for these bounds are very general and do not depend on the choice of the dominating measure they apply to both finite and infinite dimensional theta they apply in some cases where y is infinite dimensional in some cases where y is not compact in some cases where the distributions are not smooth and in some parametric cases where asymptotic normality of the posterior distribution fails

uniformly powerful goodness of fit tests the simple hypothesis is tested that the distribution of independent random variables x1 x2 xn is a given probability measure po let 7r be any sequence of partitions the alternative hypothesis is the set of probability measures p with ea ip a po a i 8 where 8 0 note the dependence of this set of alternatives on the sample size it is shown that if the effective cardinality of the partitions is of the same order as the sample size then sequences of tests exist with uniformly exponentially small probabilities of error conversely if the effective cardinality is of larger order than the sample size then no such sequence of tests exists the effective cardinality is the number of sets in the partition which exhaust all but a negligible portion of the probability under the null hypothesis

semiparametric bayesian inference in autoregressive panel data models this paper develops bayesian methods for inference in dynamic panel data models with individual effects and applies them to study longitudinal data on earnings from the panel study of income dynamics psid we study semiparametric versions of commonly used random effects autoregressive models in which the distribution of the disturbances is not restricted to fall in a parametric class to model the unknown distributions without resorting to strong parametric assumptions we draw upon recent advances in the theory and computation of nonparametric bayesian models using dirichlet process priors the overall approach can be viewed as an application of the general semiparametric bayesian approach of west muller and escobar 1994 which in turn makes use of results on bayesian density estimation by escobar 1994 and escobar and west 1995 most conventional panel data methods e g anderson and hsiao 1981 macurdy

bayesian estimation in the symmetric location problem let x i i for i 1 n where the i s are i i d f and f is symmetric about 0 f is assumed unknown or only partially known and the problem is to estimate priors are put on the pair f the priors on f are obtained from doksum s neutral to the right priors and include symmetrized dirichlet priors the marginal posterior distribution of given x1 xnis computed and its general properties studied it is found that for certain classes of distributions of the i s the posterior distribution of is for all large n a point mass at the true value of if the distribution of the i s is not exactly symmetric the bayes estimates can behave very poorly

nonparametric estimation of an instrumental regression a quasi bayesian approach based on regularized posterior we propose a quasi bayesian nonparametric approach to estimating the structural relationship among endogenous variables when instruments are available we show that the posterior distribution of is inconsistent in the frequentist sense we interpret this fact as the ill posedness of the bayesian inverse problem defined by the relation that characterizes the structural function to solve this problem we construct a regularized posterior distribution based on a tikhonov regularization of the inverse of the marginal variance of the sample which is justified by a penalized projection argument this regularized posterior distribution is consistent in the frequentist sense and its mean can be interpreted as the mean of the exact posterior distribution resulting from a gaussian prior distribution with a shrinking covariance operator

the role of beliefs in inference for rational expectations models abstract this paper discusses inference for rational expectations models estimated via minimum distance methods by characterizing the probability beliefs regarding the data generating process dgp that are compatible with given moment conditions the null hypothesis is taken to be rational expectations and the alternative hypothesis to be distorted beliefs this distorted beliefs alternative is analyzed from the perspective of a hypothetical semiparametric bayesian who believes the model and uses it to learn about the dgp this interpretation provides a different perspective on estimates test statistics and confidence regions in large samples particularly regarding the economic significance of rejections in rational expectations models a suggestive application to the equity premium puzzle is examined

some impossibility theorems in econometrics with applications to structural and dynamic models general characterizations of valid confidence sets and tests in problems which involve locally almost unidentified lau parameters are provided and applied to several econo metric models two types of inference problems are studied i inference about parame ters which are not identifiable on certain subsets of the parameter space and ii inference about parameter transformations with discontinuities when a lau parameter or parametric function has an unbounded range it is shown under general regularity conditions that any valid confidence set with level 1 a for this parameter must be unbounded with probability close to 1 a in the neighborhood of nonidentification subsets and will have a nonzero probability of being unbounded under any distribution compatible with the model no valid confidence set which is almost surely bounded does exist these properties hold even if identifying restrictions are imposed similar results also obtain for parameters with bounded ranges consequently a confidence set which does not satisfy this characterization has zero coverage probability level this will be the case in particular for wald type confidence intervals based on asymptotic standard errors furthermore wald type statistics for testing given values of a lau parameter cannot be pivotal functions i e they have distributions which depend on unknown nuisance param eters and even cannot be usefully bounded over the space of the nuisance parameters these results are applied to several econometric problems inference in simultaneous equations instrumental variables iv regressions linear regressions with autoregressive errors inference about long run multipliers and cointegrating vectors for example it is shown that standard asymptotically justified confidence intervals based on iv estimators such as two stage least squares and the associated standard errors have zero coverage probability and the corresponding t statistics have distributions which cannot be bounded by any finite set of distribution functions a result of interest for interpreting iv regressions with weak instruments furthermore expansion methods e g edgeworth expansions and bootstrap techniques cannot solve these difficulties finally in a number of cases where wald type methods are fundamentally flawed e g iv regressions with poor instruments it is observed that likelihood based methods e g likelihood ratio tests and confidence sets combined with projection techniques can easily yield valid tests and confidence sets

bayesian nonparametric estimation of the median part ii asymptotic properties of the estimates for data theta varepsilon i i 1 ldots n where varepsilon i are i i d sim f with the median of f equal to 0 but f otherwise unknown it is desired to estimate theta in doss 1985 priors are put on the pair f theta the marginal posterior distribution of theta is computed and the mean of the posterior is taken as the estimate of theta in the present paper a frequentist point of view is adopted the consistency properties of the bayes estimates computed in doss 1985 are investigated when the prior on f is of the dirichlet type any f whose median is 0 is in the support of these priors it is shown that if the varepsilon i are i i d from a discrete distribution then the bayes estimates are consistent however if the distribution of the varepsilon is is continuous the bayes estimates can be inconsistent

the relationship between damage variables and their evolution laws and microstructural and physical properties the major objective of this paper is to give a rational approach to identify the damage parameters in tensorial form based on the microstructure of the defects voids cavities microcracks and the overall physical properties of a solid material first the general representations for the effective linear and nonlinear reversible physical properties e g elastic and piezoelectric moduli of a material representative volume element rve containing a single defect of any shape are investigated second it is emphasized that the orientation distribution functions odfs of shapes separations etc of defects constitute the dominant and physically measurable signatures of the changing microstructure of a damaged material by assuming that the matrix behaves reversibly during damage nucleation and growth third it is shown that the odfs can be expanded as absolutely convergent fourier series with irreducible tensorial coefficients and that these infinitely many irreducible tensorial coefficients constitute the full list of generic damage variables furthermore to specify any given physical property only certain leading tensorial coefficients in these series are needed and therefore these leading coefficients act as the real damage tensors for such a physical property these physically based damage tensors vary significantly from one physical property to another for example it is shown that only the second and fourth tensorial coefficients effect the linear elastic moduli while only the first and third tensorial coefficients are needed in linear piezoelectricity theory finally it is observed from a simple example that the evolution equations for these physically based damage tensors normally involved tensorial terms of a higher order than those needed to define the considered physically based damage tensors consequently the evolution equations are not self closed

optimality of poisson processes intensity learning with gaussian processes in this paper we provide theoretical support for the so called sigmoidal gaussian cox process approach to learning the intensity of an inhomogeneous poisson process on a d dimensional domain this method was proposed by adams murray and mackay icml 2009 who developed a tractable computational approach and showed in simulation and real data experiments that it can work quite satisfactorily the results presented in the present paper provide theoretical underpinning of the method in particular we show how to tune the priors on the hyper parameters of the model in order for the procedure to automatically adapt to the degree of smoothness of the unknown intensity and to achieve optimal convergence rates

unbiasedness as the dual of being bayes in this note we define the notion of unbiasedness for a decision function for an arbitrary loss function this is a generalization of lehmann s 1951 definition we show that this notion of unbiasedness is a dual to the notion of being bayes that is if the role of the random variable and the parameter is interchanged then unbiasedness is equivalent to being bayes and vice versa some consequences of this fact are discussed

further thoughts on testing for causality with econometric models abstract the ontological basis for causality testing must be some empirically interpretable system leading one to specify a logically valid a priori law that can be shown to exist such paradigms as conventional linear and non linear models be they structural or time series tend to embody contradictory assumptions that unfortunately make causal interpretability problematic the paper demonstrates how these difficulties can be avoided using a stochastic coefficient approach the final part of this article is devoted to a discussion of probabilistic logic as a valid tool for scientific analysis and interpretation of causal relationships

how common is identification in parametric models abstract this paper examines whether identification is a common or rare phenomenon in nonlinear parametric models for several broad classes of models it is shown that there is an open and dense subset of identified models and that consequently the set of nonidentified models is nowhere dense the results of this paper suggest that the assumption of linearity in functional relationships to ease the conceptual and computational development of a theory can drastically limit the estimability of the model an interesting feature of this paper is that global identification is established in the context of nonlinear models