optimal critical regions for pre test estimators using a bayes risk criterion optimal critical regions for pre test estimators are developed in the context of the normal linear regression model with a conjugate prior where the criterion is bayes risk and where the pre testing involves a single coefficient when prior information is neutral regarding the sign of the coefficient it is shown that the optimal include exclude decision involves no pre testing when prior information is weighted toward a particular sign pre testing is appropriate but the optimal critical regions are quite different from those associated with the traditional pre test estimator

highest predictive density estimator in regression models abstract when the prior is only partially specified it is sometimes possible to estimate its unspecified part by maximizing the predictive density marginal likelihood of the data this paper examines the usefulness and the limitations of the priors of this kind hprd priors and the corresponding posterior mean hprd estimators in a regression setup the hprd estimator turns out to be a simple shrinkage estimator and it is found that there is an interesting relationship regarding the shrinkage factor of an estimator between the empirical bayes approach and the hprd approach the latter findings may provide some insight into getting the right size of shrinkage to attain the minimum risk the hprd prior can rarely serve as a reference prior however since the prior information it contains is too strong although this shortcoming is partly due to the nature of the hprd approach itself it can sometimes be alleviated by a more careful choice of the class of priors if the lower bound on the prior variance is set correctly the hprd estimator will provide a sensible candidate for the point estimate of the regression coefficients we also show a link between the hprd approach and the lower bound on the bayes factor

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

discontinuous skeletal gradient discretisation methods on polytopal meshes in this work we develop arbitrary order discontinuous skeletal gradient discretisations dsgd on general polytopal meshes discontinuous skeletal refers to the fact that the globally coupled unknowns are broken polynomial on the mesh skeleton the key ingredient is a high order gradient reconstruction composed of two terms i a consistent contribution obtained mimicking an integration by parts formula inside each element and ii a stabilising term for which sufficient design conditions are provided an example of stabilisation that satisfies the design conditions is proposed based on a local lifting of high order residuals on a raviart thomas nedelec subspace we prove that the novel dsgds satisfy coercivity consistency limit conformity and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems links with hybrid high order non conforming mimetic finite difference and non conforming virtual element methods are also studied numerical examples complete the exposition

efficient scores variance decompositions and monte carlo swindles abstract monte carlo swindles or variance reduction techniques exploit the experimenter s knowledge of the stochastic structure governing the simulated data to construct more precise estimates of unknown parameters alternatively one can reduce the number of replications and thus the cost needed to gain a desired level of precision this article reviews the common case of swindles based on variance decompositions for estimating efficiencies and variances of location and regression estimators we then propose a new swindle based on fisher s efficient score function that can be applied profitably to a much wider range of situations than can the gaussian over independent swindles used in many studies of robust estimators we compare these methods by performing simulations for the efficiencies of location estimates and by placing them in a simple geometric framework as an illustration the score function swindle is used to estimate the variances of pitman estimates of location for samples from selected

approximate methods for assessing influence and sensitivity in bayesian analysis we use asymptotic approximations to posterior expectations in deriving diagnostics for case influence and prior sensitivity we obtain an easily interpreted expression for the change in posterior expectation following perturbation of the problem which also produces a useful interpretation of the likelihood displacement discussed by cook weisberg in addition we propose two simple diagnostics of posterior nonnormality the leukaemia data analysed by feigl zelen are used to illustrate the diagnostic measures we discuss

consistent sets of estimates for regressions with correlated or uncorrelated measurement errors in arbitrary subsets of all variables over the last decade the problem of measurement errors in the independent variables of a regression equation has attracted renewed interest among econometricians in the fifties and sixties the problem was considered to be more or less hopeless due to its inherent underidentification e g theil 1971 apart from instrumental variables the most frequently cited textbook solution was wald s method of grouping wald 1940 recent insight into the properties of the method of grouping can be interpreted as making this method worthless in most practical cases pakes 1982 since about 1970 new approaches to the problem have been explored basically along three lines viz embedding the error ridden equation into a set of multiple equations e g zellner 1970 goldberger 1972 into a set of simultaneous equations e g hsiao 1976 geraci 1976 and using the dynamics of the equation if present e g maravall and aigner 1977 in view of the underidentification of the basic model it is clear that all these methods invoke additional information of some kind if this information takes the form of exact or stochastic knowledge of certain parameters in the model the construction of consistent estimators is fairly straightforward e g fuller 1980 kapteyn and wansbeek 1984 for an overview of the state of the art see aigner et al 1984 an approach somewhat orthogonal to the ones described above has been to take the model as it is and to use prior ideas about the size of the measurement errors to diagnose how serious the probem is examples are blomqvist 1972 hodges and moore 1972 and davies and hutton 1975 leamer 1983 starts from the opposite direction by asking how serious the measurement error problem has to be in order to render the data useless for inference that is to say when measurement error is large enough to make it impossible to put bounds on regression parameters in an empirical example he shows that even very small measurement errors in some explanatory variables would open up the possibility of perfectly collinear explanatory variables and hence make the data useless for statistical inference at least without additional prior information the most systematic analysis of the information loss caused by measurement error is due to klepper and leamer 1984 they start out by invoking a minimal amount of prior information and then ask the question under what conditions it is still possible to make some inferences regarding the vector of unknown regression parameters p in the special case where the measurement errors are assumed uncorrelated and the k 1 estimates of 3 obtained by regressing each of the k 1 variables involved i e the one dependent variable and the k independent variables on the remaining k variables are all in the same orthant one can bound the ml estimates of p in that case the convex hull of the k 1 regressions contains all possible ml estimates and any point in the hull is a possible ml estimate if the k 1 regressions are not all in the same orthant then the set of ml estimates is unbounded in that case klepper and leamer 1984 introduce extra prior information which allows them to bound the set of maximum likelihood estimates the prior information comes in two forms firstly a researcher is supposed to be able to specify a maximum value of r2 if all exogenous variables were measured accurately it is shown that if this maximum is low enough one can again bound the set of ml estimates by a convex hull secondly if

ridge regression estimation of the rotterdam model abstract the best guesses of unknown coefficients specified in theil s model of introspection are like predictions and not like de finetti s prevision and therefore not the values taken by random variables constrained least squares procedures can be formulated which are free of these difficulties the ridge estimator is a simple version of a constrained least squares estimator which can be made operational even when little prior information is available our operational ridge estimators are nearly minimax and are not less stable than least squares in the presence of high multicollinearity finally we have presented the ridge estimates for the rotterdam demand model

bounding posterior means by model criticism abstract to handle the sensitivity of posterior inferences to prior choices leamer 1983 suggests that one specify a class of priors that is large enough to include all possible priors and then derive the set of posterior distributions associated with it in many practical situations however this set of posteriors turns out to be too large to be useful even with informative data frequently the culprit is those dogmatic priors that are not compatible with the data information and removal of such priors often significantly narrows down the set of posteriors when a complete elicitation process requires excessive time and resources a feasible alternative is to use the sample information to help the researcher identify unreasonable priors the approach proposed in this paper is to bound the posterior means of the regression parameters by model criticism in the sense of box 1980 more specifically it restricts priors to those for which the predictive density of the observation is not too small a set of posteriors is then obtained for this restricted class of priors the proposed procedure will be applied to u s money stock data to examine the behavior of the proposed bounds

analysis of variable degree hdg methods for convection diffusion equations part ii semimatching nonconforming meshes in this paper we provide a projection based analysis of the hversion of the hybridizable discontinuous galerkin methods for convectiondiffusion equations on semimatching nonconforming meshes made of simplexes the degrees of the piecewise polynomials are allowed to vary from element to element we show that for approximations of degree k on all elements the order of convergence of the error in the diffusive flux is k 1 and that that of a projection of the error in the scalar unknown is 1 for k 0 and k 2 for k 0 we also show that for the variable degree case the projection of the error in the scalar variable is h times the projection of the error in the vector variable provided a simple condition is satisfied for the choice of the degree of the approximation on the elements with hanging nodes these results hold for any bounded irregularity index of the nonconformity of the mesh moreover our analysis can be extended to hypercubes

sample size determination for robust bayesian analysis this article considers a robust bayesian approach to the sample size determination problem we focus on global bayesian robustness that studies lower bound ln upper bound un and range rn of posterior quantities of interest obtained as the prior varies in a class of distributions specifically we are interested in the selection of an appropriate sample size that gives guarantees to the researcher of observing a small value of the range and depending on the problems either a sufficiently large lower bound or a sufficiently small upper bound toward this end we approach the problem as a design issue and provide new sample size determination criteria based on summaries of the predictive distributions of ln un and rn such as expectations and tail probabilities relationships and comparison to standard classical and nonrobust bayesian methods are discussed the proposed methods are studied for the normal model with conjugate priors and used for choosing the size of a clinical trial

coordinate free ridge regression bounds abstract the dependence of ridge regression estimates on the choice of parameterization is defined with reference to the hull of all possible ridge estimates for all possible parameterizations a reanalysis of hoerl and kennard s 1970 10 factor example and marquardt and snee s 1975 acetylene data reveals that most conclusions depend on the parameterization

a priori and a posterior error estimate of new weak galerkin finite element methods for second order elliptic interface problems on polygonal meshes abstract in this paper we present a posteriori error estimate of a new weak galerkin wg finite element method for the second order elliptic interface problems instead of introducing the local weak derivatives on each element the bilinear form is computed through the inner product of traditional defined derivatives the proposed numerical scheme is symmetric and positive definite and can be applied to polygonal meshes in the a posteriori error estimate we construct a error indicator that can be applied to polygonal meshes or meshes with hanging nodes the reliability and efficiency of the designed error estimator have been proved in this work extensive numerical tests are performed to validate our algorithm these results demonstrate the effectiveness of the adaptive mesh refinement guided by the proposed error estimator

an inverse problem of reflection abstract this paper developed an analytic method to solve the inverse problem of reflection the direct problem is to determine the reflection at the top of a known stratified medium bounded below by a reflector the inverse problem is to determine the reflection of a reflector from the measured reflection at the top of the medium the main results are based on the iteration of solutions of two initial value differential equations they are suitable for numerical computation a mathematical model is constructed on an abstract space for broad applications which include radiative transfer with inhomogeneous and anisotropic scattering the solution is general is not unique a condition has been established for a unique solution non unique solutions are also constructed

a weak galerkin finite element method for the navier stokes equations abstract this paper introduces a weak galerkin wg finite element method for the navier stokes equations in the primal velocity pressure formulation optimal order error estimates are established for the corresponding numerical approximations it must be emphasized that the wg finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular numerical experiments are presented to support the theoretical results

bayes theorem for choquet capacities we give an upper bound for the posterior probability of a measurable set a when the prior lies in a class of probability measures p the bound is a rational function of two choquet integrals if p is weakly compact and is closed with respect to majorization then the bound is sharp if and only if the upper prior probability is 2 alternating the result is used to compute bounds for several sets of priors used in robust bayesian inference the result may be regarded as a characterization of 2 alternating choquet capacities

discontinuous hp finite element methods for advection diffusion reaction problems we consider the hp version of the discontinuous galerkin finite element method dgfem for second order partial differential equations with nonnegative characteristic form this class of equations includes second order elliptic and parabolic equations advection reaction equations as well as problems of mixed hyperbolic elliptic parabolic type our main concern is the error analysis of the method in the absence of streamline diffusion stabilization in the hyperbolic case an hp optimal error bound is derived here we consider only advection reaction problems which satisfy a certain standard positivity condition in the self adjoint elliptic case an error bound that is h optimal and p suboptimal by frac 1 2 a power of p is obtained these estimates are then combined to deduce an error bound in the general case for elementwise analytic solutions the method exhibits exponential rates of convergence under p refinement the theoretical results are illustrated by numerical experiments

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

a hybridizable discontinuous galerkin method for steady state convection diffusion reaction problems in this article we propose a novel discontinuous galerkin method for convection diffusion reaction problems characterized by three main properties the first is that the method is hybridizable this renders it efficiently implementable and competitive with the main existing methods for these problems the second is that when the method uses polynomial approximations of the same degree for both the total flux and the scalar variable optimal convergence properties are obtained for both variables this is in sharp contrast with all other discontinuous methods for this problem the third is that the method exhibits superconvergence properties of the approximation to the scalar variable this allows us to postprocess the approximation in an element by element fashion to obtain another approximation to the scalar variable which converges faster than the original one in this paper we focus on the efficient implementation of the method and on the validation of its computational performance with this aim e

implementation of discontinuous skeletal methods on arbitrary dimensional polytopal meshes using generic programming discontinuous skeletal methods approximate the solution of boundary value problems by attaching discrete unknowns to mesh faces hence the term skeletal while allowing these discrete unknowns to be chosen independently on each mesh face hence the term discontinuous cell based unknowns which can be eliminated locally by a schur complement technique also known as static condensation are also used in the formulation salient examples of high order discontinuous skeletal methods are hybridizable discontinuous galerkin methods and the recently devised hybrid high order methods some major benefits of discontinuous skeletal methods are that their construction is dimension independent and that they offer the possibility to use general meshes with polytopal cells and non matching interfaces in this work we show how this mathematical flexibility can be efficiently replicated in a numerical software using generic programming we describe a number of generic algorithms and data structures for high order discontinuous skeletal methods within a write once run on any kind of mesh framework the computational efficiency of the implementation is assessed on the poisson model problem discretized using various polytopal meshes and the hybrid high order method

a hybrid high order method for a coupled stokes darcy problem on general meshes abstract in this work a hybrid high order hho method on general meshes is presented to solve a coupled stokes darcy problem with the beavers joseph saffman interface condition constructed on polynomials of arbitrary degree k 0 the numerical method is established in terms of discrete unknowns attached to mesh faces and cells or elements the unified discrete scheme for stokes equation and darcy equation is given by the continuity condition of the interface the unique solvability of the discrete scheme is proved moreover the energy error estimate for the velocity and l 2 error estimate for pressure of order k 1 are derived finally a series of numerical experiments are reported to illustrate the accuracy mass conservation and robustness of our method

ranges of posterior probabilities for quasiunimodal priors with specified quantiles abstract suppose several quantiles of the prior distribution for are specified or equivalently the prior probabilities of a partitioning collection of intervals ii are given in addition suppose that the prior distribution is assumed to be unimodal rather than selecting a single prior distribution to perform a bayesian analysis it is of interest to consider the class of all prior distributions compatible with these inputs for this class and unimodal likelihood functions the ranges of the posterior probabilities of the ii and the ranges of the posterior cdf at the specified prior quantiles were determined in berger and o hagan in press unfortunately calculations with this class can be difficult here a similar much more easily analyzed class of quasiunimodal prior distributions is considered and compared with other classes

scattering operators and propagation solutions for a non stationary transfer equation abstract physically meaningful mild scattering operator solutions for non stationary radiative transfer equations including diffuse reflection and transmission by finite optical thickness plane parallel or discrete angular redistribution curved geometries are presented our method is based on the construction of scattering matrices from a c 0 propagation group the difficulty of this problem is the unbounded generator occurring in the non stationary case in an energy dissipative case the local existence of scattering operators is extended to global existence

sensitivity in bayesian statistics the prior and the likelihood abstract one paradigm for sensitivity analyses in bayesian statistics is to specify a reasonable class of priors and to compute the corresponding class of posterior inferences the class is chosen to represent uncertainty about the prior there is often additional uncertainty however about the family of sampling distributions this article introduces a method for computing ranges of posterior expectations over reasonable classes of sampling distributions that lie close to a given parametric family by treating the prior as a probability measure on the space of sampling distributions this article also gives a unified treatment to what are usually considered two separate problems sensitivity to the prior and sensitivity to the sampling model first the notion of close to is made explicit then an algorithm is given for turning ratio linear problems into sequences of linear problems in addition to solving the problem at hand the algorithm simplifies many other robust bayesian computational pro

errors in variables in linear systems this paper extends the simple errors in variable bound to the setting of systems of equations both diagonal and nondiagonal measurement error covariance matrices are considered in the nondiagonal case the analogue of the simple errors in variable interval of estimates is an ellipsoid with diagonal equal to the line segment connecting the direct least squares with a two stage least squares estimate for the diagonal case the set of estimates under some conditions must lie within the convex hull of 2k points

robust bayesian analysis of a multivariate dynamic model abstract recently a bayesian approach has been widely used to infer parameters of complex models from finite sample data in business finance social and physical sciences bayesian applications in finance literature are growing due to their exposure to prior beliefs that macroeconomic variables influence the financial behavior of customers investors and firm s performance this paper employs a robust bayesian analysis model to capture a complex multi dimensional process and estimate the required parameters of interest it considers a class of prior distributions for the parameters that is a mixture of natural conjugate base priors and a contamination class of prior distributions the conditional type ii maximum likelihood posterior densities and posterior means for the coefficients vector and autoregressive parameter are derived for the coefficients vector and autoregressive parameter of interest the markov chain monte carlo mcmc method has been used for obtaining the marginal posterior densities and bayes estimators for various combinations of the parameters the usefulness of the model is demonstrated on indian financial firms balance sheets data between 1992 to 2015 to observe the sensitivity of the proposed model the result shows most of the corporate performance variables are significantly affecting the firm s profitability with the exception of interest coverage ratio

a discontinuous skeletal method for the viscosity dependent stokes problem we devise and analyze arbitrary order nonconforming methods for the discretization of the viscosity dependent stokes equations on simplicial meshes we keep track explicitly of the viscosity and aim at pressure robust schemes that can deal with the practically relevant case of body forces with large curl free part in a way that the discrete velocity error is not spoiled by large pressures the method is inspired from the recent hybrid high order hho methods for linear elasticity after elimination of the auxiliary variables by static condensation the linear system to be solved involves only discrete face based velocities which are polynomials of degree k ge 0 and cell wise constant pressures our main result is a pressure independent energy error estimate on the velocity of order k 1 the main ingredient to achieve pressure independence is the use of a divergence preserving velocity reconstruction operator in the discretization of the body forces we also prove an l 2 pressure estimate of order k 1 and an l 2 velocity estimate of order k 2 the latter under elliptic regularity the local mass and momentum conservation properties of the discretization are also established finally two and three dimensional numerical results are presented to support the analysis

a discontinuous galerkin method with lagrange multipliers for spatially dependent advection diffusion problems abstract a higher accuracy discontinuous galerkin method with lagrange multipliers dglm is presented for the solution of the advection diffusion equation with a spatially varying advection field in the high peclet number regime where the classical polynomial finite element method fem produces spurious oscillations in the solution at practical mesh resolutions the proposed dglm method is based on discontinuous polynomial shape functions that are attached to an element rather than its nodes it overcomes the aforementioned spurious oscillation issue by enriching these functions with approximate free space solutions of homogeneous equations derived from an asymptotic analysis of the governing partial differential equation inspired by prandtl s boundary layer theory these enrichment functions are capable of resolving exponential parabolic and corner boundary layers at relatively coarse mesh resolutions the proposed method enforces a weak continuity of the solution approximation across inter element boundaries using polynomial lagrange multipliers which makes it a hybrid method however unlike other hybrid methods it operates directly on the second order form of the advection diffusion equation and does not require any stabilization its intrinsic performance and its superiority over the higher order polynomial fem are demonstrated for several test problems at peclet numbers ranging from one thousand to one billion

searchlight on a target with diffuse background i a mathematical model of a searchlight on a target with diffused background is constructed based on three dimensional radiative transfer and integral operators the main result is the recovery of the true target reflection from the measured reflection which consists of background light and multiscattering between atmospheric layer and the earth surface an exact solution is solved and approximations are obtained by use of the assumption that the searchlight is strong and focused and the background light is weak and uniform in space in a given narrow frequency band of intensity

an extension of the crouzeix raviart space to general meshes with application to quasi incompressible linear elasticity and stokes flow in this work we introduce a discrete functional space on general polygonal or polyhedral meshes which mimics two important properties of the standard crouzeix raviart space namely the continuity of mean values at interfaces and the existence of an interpolator which preserves the mean value of the gradient inside each element the construction borrows ideas from both cell centered galerkin and hybrid finite volume methods the discrete function space is defined from cell and face unknowns by introducing a suitable piecewise affine reconstruction on a fictitious pyramidal subdivision of the original mesh two applications are considered in which the discrete space plays an important role namely i the design of a locking free primal as opposed to mixed method for quasi incompressible planar elasticity on general polygonal meshes ii the design of an inf sup stable method for the stokes equations on general polygonal or polyhedral meshes in this context we also propose a general modification applicable to any suitable discretization which guarantees that the velocity approximation is unaffected by the presence of large irrotational body forces provided a helmholtz decomposition of the right hand side is available the relation between the proposed methods and classical finite volume and finite element schemes on standard meshes is investigated finally similar ideas are exploited to mimic key properties of the lowest order raviart thomas space on general polygonal or polyhedral meshes

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

reconciling bayesian and frequentist evidence in the one sided testing problem abstract for the one sided hypothesis testing problem it is shown that it is possible to reconcile bayesian evidence against h 0 expressed in terms of the posterior probability that h 0 is true with frequentist evidence against h 0 expressed in terms of the p value in fact for many classes of prior distributions it is shown that the infimum of the bayesian posterior probability of h 0 is equal to the p value in other cases the infimum is less than the p value the results are in contrast to recent work of berger and sellke 1987 in the two sided point null case where it was found that the p value is much smaller than the bayesian infimum some comments on the point null problem are also given

a discontinuous skeletal method for advection diffusion reaction on general meshes we design and analyze an approximation method for advection diffusion reaction equa tions where the generalized degrees of freedom are polynomials of order k 0 at mesh faces the method hinges on local discrete reconstruction operators for the diffusive and advective derivatives and a weak enforcement of boundary conditions fairly general meshes with poly topal and nonmatching cells are supported arbitrary polynomial orders can be considered including the case k 0 which is closely related to mimetic finite difference mixed hybrid finite volume methods the error analysis covers the full range of peclet numbers including the delicate case of local degeneracy where diffusion vanishes on a strict subset of the domain computational costs remain moderate since the use of face unknowns leads to a compact stencil with reduced communications numerical results are presented

a hybrid high order method for a class of quasi newtonian stokes equations on general meshes abstract in this paper we introduce a hybrid high order hho discrete scheme for numerically solving a class of incompressible quasi newtonian stokes equations in r 2 the presented hho method depends on hybrid discrete velocity unknowns at cells and edges and pressure unknowns at cells benefiting from the hybridization of unknowns the computation cost can be reduced by the technique of static condensation and the solvability of the static condensation algebra system is proved furthermore we study the hho scheme by polynomials of arbitrary degrees k k 1 on the general meshes and geometries the unique solvability of the discrete scheme is proved additionally the optimal a priori error estimates for the velocity gradient and pressure approximations are obtained finally we provide several numerical results to verify the good performance of the proposed hho scheme and confirm the optimal approximation properties on a variety of meshes and geometries

testing a point null hypothesis the irreconcilability of p values and evidence abstract the problem of testing a point null hypothesis or a small interval null hypothesis is considered of interest is the relationship between the p value or observed significance level and conditional and bayesian measures of evidence against the null hypothesis although one might presume that a small p value indicates the presence of strong evidence against the null such is not necessarily the case expanding on earlier work especially edwards lindman and savage 1963 and dickey 1977 it is shown that actual evidence against a null as measured say by posterior probability or comparative likelihood can differ by an order of magnitude from the p value for instance data that yield a p value of 05 when testing a normal mean result in a posterior probability of the null of at least 30 for any objective prior distribution objective here means that equal prior weight is given the two hypotheses and that the prior is symmetric and nonincreasing away from the null other definiti

ranges of posterior measures for priors with unimodal contaminations we consider the problem of robustness or sensitivity of given bayesian posterior criteria to specification of the prior distribution criteria considered include the posterior mean variance and probability of a set for credible regions and hypothesis testing uncertainty in an elicited prior go is modelled by an e contamination class f x 1 c 1t0 eq q e q where reflects the amount of probabilistic uncertainty in g and q is a class

bayes like decision making with upper and lower probabilities abstract we consider the use of interval valued probabilities to represent the support lent to the hypothesis that the parameter value lies in a subset a of the parameter set when we observe x know the likelihoods f e and have some prior information concerning the parameter our model for prior information is that of a salient prior distribution in which we have little confidence although we have much less confidence in any alternative prior we consider notions of acceptable and coherent decision making as well as notions of being able to achieve a bayes rule and least commitment throughout we are motivated to preserve some of the elements of bayesian decision making without thereby committing ourselves to unwarranted claims of knowledge

discontinuous galerkin methods for anisotropic semidefinite diffusion with advection we construct and analyze a discontinuous galerkin method to solve advection diffusion reaction pdes with anisotropic and semidefinite diffusion the method is designed to automatically detect the so called elliptic hyperbolic interface on fitted meshes the key idea is to use consistent weighted average and jump operators optimal estimates in the broken graph norm are proven these are consistent with well known results when the problem is either hyperbolic or uniformly elliptic the theoretical results are supported by numerical evidence

reduced form estimation and prediction from uncertain structural models a generic approach abstract this paper proposes a direct hierarchical model for the analysis of the reduced forms rf of the classical linear simultaneous equations models sem the mean of the prior p d f on rf coefficients is partially assessed with the aid of the traditionally exact a priori restrictions its variance may be partially specified to reflect degrees of model uncertainty and reliability of inferences that are asymptotically valid bayes or maximum joint probability estimates are derived for both the rf and the structural parameters one variant of the rf estimator is shown to have desirable limiting and finite sample properties it performs well in an empirical example which incorporates some elements of the two real life problems of concern to us simultaneous presence of different types of misspecification and small sample sizes

s values conventional context minimal measures of the sturdiness of regression coefficients this paper proposes a context minimal range of alternative regression models that is used to generate a range of alternative estimates a prior distribution is assumed with a zero mean but an ambiguous covariance matrix the choice of the prior covariance matrix is facilitated by transformation to standardized variables which makes the prior expected r2 equal to the sum of the prior variances three different ranges of the prior expected r2 are used to define three different intervals of prior covariance matrices which are used to produce three different sets of s values

a hybrid high order method for the incompressible navier stokes equations based on temam s device abstract in this work we propose a novel hybrid high order method for the incompressible navier stokes equations based on a formulation of the convective term including temam s device for stability the proposed method has several advantageous features it supports arbitrary approximation orders on general meshes including polyhedral elements and non matching interfaces it is inf sup stable it is locally conservative it supports both the weak and strong enforcement of velocity boundary conditions it is amenable to efficient computer implementations where a large subset of the unknowns is eliminated by solving local problems inside each element particular care is devoted to the design of the convective trilinear form which mimicks at the discrete level the non dissipation property of the continuous one the possibility to add a convective stabilisation term is also contemplated and a formulation covering various classical options is discussed the proposed method is theoretically analysed and an energy error estimate in h k 1 with h denoting the meshsize is proved under the usual data smallness assumption a thorough numerical validation on two and three dimensional test cases is provided both to confirm the theoretical convergence rates and to assess the method in more physical configurations including in particular the well known two and three dimensional lid driven cavity problems

a hybrid high order locking free method for linear elasticity on general meshes abstract we devise an arbitrary order locking free method for linear elasticity the method relies on a pure displacement primal formulation and leads to a symmetric positive definite system matrix with compact stencil the degrees of freedom are vector valued polynomials of arbitrary order k 1 on the mesh faces so that in three space dimensions the lowest order scheme only requires 9 degrees of freedom per mesh face the method can be deployed on general polyhedral meshes the key idea is to reconstruct the symmetric gradient and divergence inside each mesh cell in terms of the degrees of freedom by solving inexpensive local problems the discrete problem is assembled cell wise using these operators and a high order stabilization bilinear form locking free error estimates are derived for the energy norm and for the l 2 norm of the displacement with optimal convergence rates of order k 1 and k 2 respectively for smooth solutions on general meshes the theoretical results are confirmed numerically and the cpu cost is evaluated on both standard and polygonal meshes

robust bayesian experimental designs in normal linear models we address the problem of finding a design that minimizes the bayes risk with respect to a fixed prior subject to being robust with respect to misspecification of the prior uncertainty in the prior is formulated in terms of having a family of priors instead of one single prior two different classes of priors are considered f1 is a family of conjugate priors and a second family of priors f2 is induced by a metric on the space of nonnegative measures the family f1 has earlier been suggested by leamer and polasek while f2 was considered by derobertis and hartigan and berger the setup assumed is that of a canonical normal linear model with independent homoscedastic errors optimal robust designs are considered for the problem of estimating the vector of regression coefficients or a linear combination of the regression coefficients and also for testing and set estimation problems concrete examples are given for polynomial regression and completely randomized designs a very surprising finding is that for r2 the same design is optimal for a variety of different problems with different loss structures in general the results for f2 are significantly more substantive our results are applicable to group decision making and reconciliation of opinions among experts with different priors

bayesian robustness and the stein effect abstract in simultaneous estimation of normal means it is shown that through use of the stein effect surprisingly large gains of a bayesian nature can be achieved at little or no cost if the prior information is misspecified this provides a justification in terms of robustness with respect to mis specification of the prior for employing the stein effect even when combining a priori independent problems i e problems in which no empirical bayes effects are obtainable to study this issue a class of minimax estimators that closely mimic the conjugate prior bayes estimators is introduced

an adaptive empirical bayes estimator of the multivariate normal mean under quadratic loss abstract for shrinkage estimators to achieve significant risk improvement over their traditional competitors one must identify the region or subspace where the unknown location vector lies or is thought likely to lie a priori when vague or conflicting priors suggest that a broad class of estimators may be effective under a squared error loss measure new minimax or near minimax empirical bayes estimators are proposed that make use of stein s unbiased estimator of the risk to identify the optimum risk effective shrinkage estimator

numerical approximation of poroelasticity with random coefficients using polynomial chaos and hybrid high order methods abstract in this work we consider the biot problem with uncertain poroelastic coefficients the uncertainty is modeled using a finite set of parameters with prescribed probability distribution we present the variational formulation of the stochastic partial differential system and establish its well posedness we then discuss the approximation of the parameter dependent problem by non intrusive techniques based on polynomial chaos decompositions we specifically focus on sparse spectral projection methods which essentially amount to performing an ensemble of deterministic model simulations to estimate the expansion coefficients the deterministic solver is based on a hybrid high order discretization supporting general polyhedral meshes and arbitrary approximation orders we numerically investigate the convergence of the probability error of the polynomial chaos approximation with respect to the level of the sparse grid finally we assess the propagation of the input uncertainty onto the solution considering an injection extraction problem

bayesian minimax estimation of the normal model with incomplete prior covariance matrix specification this work addresses the issue of bayesian robustness in the multivariate normal model when the prior covariance matrix is not completely specified but rather is described in terms of positive semi definite bounds this occurs in situations where for example the only prior information available is the bound on the diagonal of the covariance matrix derived from some physical constraints and that the covariance matrix is positive semi definite but otherwise arbitrary under the conditional gamma minimax principle previous work by dasgupta and studden shows that an analytically exact solution is readily available for a special case where the bound difference is a scaled identity the goal in this work is to consider this problem for general positive definite matrices the contribution in this paper is a theoretical study of the geometry of the minimax problem extension of previous results to a more general case is shown and a practical algorithm that relies on semi definite programming and the convexity of the minimax formulation is derived although the algorithm is numerically exact for up to the bivariate case its exactness for other cases remains open numerical studies demonstrate the accuracy of the proposed algorithm and the robustness of the minimax solution relative to standard and recently proposed methods

robust bayes and empirical bayes analysis with _ epsilon contaminated priors on considere deux directions dans l analyse bayesienne robuste la premiere consiste a determiner l etendue des probabilites a posteriori d un ensemble la seconde consiste a selectionner une bonne distribution a priori

global sensitivity results for generalized least squares estimates abstract the covariance matrix for the residuals of a regression process is written as the identity matrix plus a matrix v the matrix v is bounded from above and the corresponding set of generalized least squares estimates is identified the extreme estimates in this set are functions of the usual t statistics in particular the number t k 8 1 2 t measures the influence of reweighting extreme observations where t k gives the degrees of freedom t is the t statistic and the weights on observations are allowed to vary by a factor of at most 2

an approach to robust bayesian analysis for multidimensional parameter spaces abstract in some multiparameter bayesian problems the prior is more naturally described as a joint density for all of the parameters and in others it is more naturally described as a product of marginal and conditional densities although ordinary quantile e contamination and density ratio classes of priors may be sensible choices for robust bayesian analyses in the former case they may be less satisfactory in the latter this article shows how to construct product classes of priors for use in the second case using one class of priors for the marginal and another for each conditional then the article introduces density bounded classes as an alternative to the more familiar classes for representing uncertainty about the prior it is often of interest to find tight bounds on posterior expectations as the prior ranges over the class of priors an apparently difficult nonlinear optimization problem regardless of which class of priors is used this article describes a technique for turning the problem into

a hybrid high order discretisation of the brinkman problem robust in the darcy and stokes limits abstract in this work we develop and analyse a novel hybrid high order discretisation of the brinkman problem the method hinges on hybrid discrete velocity unknowns at faces and elements and on discontinuous pressures based on the discrete unknowns we reconstruct inside each element a stokes velocity one degree higher than face unknowns and a darcy velocity in the raviart thomas nedelec space these reconstructed velocities are respectively used to formulate the discrete versions of the stokes and darcy terms in the momentum equation along with suitably designed penalty contributions the proposed construction is tailored to yield optimal error estimates that are robust throughout the entire spectrum of local stokes or darcy dominated regimes as identified by a dimensionless number which can be interpreted as a friction coefficient the singular limit corresponding to the darcy equation is also fully supported by the method numerical examples corroborate the theoretical results this paper also contains two contributions whose interest goes beyond the specific method and application treated in this work an investigation of the dependence of the constant in the second korn inequality on star shaped domains and its application to the study of the approximation properties of the strain projector in general sobolev seminorms

robust filtering through coherent lower previsions the classical filtering problem is re examined to take into account imprecision in the knowledge about the probabilistic relationships involved imprecision is modeled in this paper by closed convex sets of probabilities we derive a solution of the state estimation problem under such a framework that is very general it can deal with any closed convex set of probability distributions used to characterize uncertainty in the prior likelihood and state transition models this is made possible by formulating the theory directly in terms of coherent lower previsions that is of the lower envelopes of the expectations obtained from the set of distributions the general solution is specialized to two particular classes of coherent lower previsions the first consists of a family of gaussian distributions whose means are only known to belong to an interval the second is the so called linear vacuous mixture model which is a family made of convex combinations of a known nominal distribution e g a gaussian with arbitrary distributions for the latter case we empirically compare the proposed estimator with the kalman filter this shows that our solution is more robust to the presence of modelling errors in the system and that hence appears to be a more realistic approach than the kalman filter in such a case

a high order discontinuous galerkin method for unsteady advection diffusion problems abstract a high order discontinuous galerkin method with lagrange multipliers is presented for the solution of unsteady advection diffusion problems in the high peclet number regime it operates directly on the second order form of the governing equation and does not require any stabilization its spatial basis functions are chosen among the free space solutions of the homogeneous form of the partial differential equation obtained after time discretization it also features lagrange multipliers for enforcing a weak continuity of the approximated solution across the element interface boundaries this leads to a system of differential algebraic equations which are time integrated by an implicit family of schemes the numerical stability of these schemes and the well posedness of the overall discretization method are supported by a theoretical analysis the performance of this method is demonstrated for various high peclet number constant coefficient model flow problems

bayesian variants of some classical semiparametric regression techniques abstract this paper develops new bayesian methods for semiparametric inference in the partial linear normal regression model y z f x e where f is an unknown function these methods draw solely on the normal linear regression model with natural conjugate prior hence posterior results are available which do not suffer from some problems which plague the existing literature such as computational complexity methods for testing parametric regression models against semiparametric alternatives are developed we discuss how these methods can at some cost in terms of computational complexity be extended to other models e g qualitative choice models or those involving censoring or truncation and provide precise details for a semiparametric probit model we show how the assumption of normal errors can easily be relaxed

weak galerkin based a posteriori error estimates for second order elliptic interface problems on polygonal meshes abstract in this paper we present a posteriori error estimate of weak galerkin wg finite element methods based on the second order elliptic interface problems this estimate can be applied to polygonal meshes or meshes with hanging nodes the reliability and efficiency of the designed error estimator has been proved in this work extensive numerical tests are performed to validate our algorithm these results demonstrate the effectiveness of the adaptive mesh refinement guided by the proposed error estimator

estimation problems in an input and output system abstract we investigate smooth and filtered estimation problems in an input and output system the hamiltonian system is decomposed into internal and external states with this approach we are able to obtain general results which include some well known ones

robust bayesian sample size determination in clinical trials this article deals with determination of a sample size that guarantees the success of a trial we follow a bayesian approach and we say an experiment is successful if it yields a large posterior probability that an unknown parameter of interest an unknown treatment effect or an effects difference is greater than a chosen threshold in this context a straightforward sample size criterion is to select the minimal number of observations so that the predictive probability of a successful trial is sufficiently large in the paper we address the most typical criticism to bayesian methods their sensitivity to prior assumptions by proposing a robust version of this sample size criterion specifically instead of a single distribution we consider a class of plausible priors for the parameter of interest robust sample sizes are then selected by looking at the predictive distribution of the lower bound of the posterior probability that the unknown parameter is greater than a chosen threshold for their flexibility and mathematical tractability we consider classes of epsilon contamination priors as specific applications we consider sample size determination for a phase iii trial

bounding parameters in a linear regression model with a mismeasured regressor using additional information this paper discusses a linear regression model with a mismeasured regressor in which the measurement error is correlated with both the latent variable and the regression error we use a linear structure to capture the correlation between the measurement error and the latent variable this paper shows that the variance of the latent variable is very useful for revealing information on the parameters which otherwise cannot be obtained with such a nonclassical measurement error the main result is that the finite bounds on the parameters can be found using the variance of the latent variable regardless of how severely the measurement error and the regression error are correlated if the mismeasured regressor contains enough information on the latent one this paper also discusses the special but interesting case of the latent variable being dichotomous in this case the mean of the latent variable may even reveal information on the correlation between the measurement error and the regression error all the bounds developed in the paper are tight