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

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

contains and inside relationships within combinatorial pyramids irregular pyramids are made of a stack of successively reduced graphs embedded in the plane such pyramids are used within the segmentation framework to encode a hierarchy of partitions the different graph models used within the irregular pyramid framework encode different types of relationships between regions this paper compares different graph models used within the irregular pyramid framework according to a set of relationships between regions we also define a new algorithm based on a pyramid of combinatorial maps which allows to determine if one region contains the other using only local calculus

approximations of marginal tail probabilities and inference for scalar parameters summary in many situations inference for a scalar parameter in the presence of nuisance parameters requires integration of either a joint density of pivotal quantities or a joint posterior density for such inference accurate approximations of marginal tail probabilities are useful to avoid high dimensional integrals two tail probability approximations are developed in this paper numerical results given for conditional inference in location scale and linear regression models show the approximations to be generally accurate even for small sample sizes

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

saddlepoint approximations for regression models summary this paper uses the techniques of saddlepoint or tilted exponential approximation to develop an approximation to the small sample distribution of estimators defined by a system of estimating equations when observations are independently but not identically distributed this allows for the explicit treatment of models with explanatory variables the approximation is applied to l1 regression and the accuracy of the approximation is demonstrated in a case in which the usual normal approximation is very poor an analysis of the example indicates the usefulness of a new variant of exponential tilting in calculating the transformation measure a quantity which plays an important role in assuring the accuracy of the approximation

asymptotic normality with small relative errors of posterior probabilities of half spaces some examples of half spaces of interest in parameter spaces are in a clinical trial of a treatment versus placebo the half spaces where the treatment is a helpful or b harmful thus one may not only want to test the hypothesis that the treatment c makes no difference but to assign posterior probabilities to a b and c under conditions as unrestrictive as possible on the choice of prior probabilities e g dudley and haughton 2001 more generally we have in mind applications to model selections as in the bic criterion of schwarz 1978 and its extensions poskitt 1987 haughton 1988 specifically to one sided models and multiple data sets dudley and haughton 1997

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

commuting finite blaschke products with no fixed points in the unit disk abstract in this paper we study when two finite blaschke products commute we complete previous results by chalendar and mortini when they have a fixed point in the unit disk and by arteaga when they do not have a fixed point in the unit disk

stable and invariant adjusted directed likelihoods summary for the purpose of making inferences for a one dimensional interest parameter or constructing approximate complementary ancillaries or residuals the directed likelihood or signed square root of the likelihood ratio statistic can be adjusted so that the resulting modified directed likelihood is under ordinary repeated sampling approximately standard normal with error of o n 312 conditional on a suitable ancillary statistic and hence unconditionally in general suitable specification of the ancillary statistic may be difficult we introduce two adjusted directed likelihoods which are similar to the modified directed likelihood but do not require the specification of the ancillary statistic the error of the standard normal approximation to the distribution of these new adjusted directed likelihoods is o n 1 conditional on any reasonable ancillary statistic which is still an improvement over the unadjusted directed likelihoods

a laplace approximation to the moments of a ratio of quadratic forms summary the laplace method for approximating integrals is applied to give a general approximation for the kth moment of a ratio of quadratic forms in random variables the technique utilises the existence of a dominating peak at the boundary point on the range of integration as closed form and tractable formulae do not exist in general this simple approximation which only entails basic algebraic operations has evident practical appeal we exploit the approximation to provide an approximate mean bias function for the least squares estimator of the coefficient of the lag dependent variable in a first order stochastic difference equation

laplace approximations for posterior expectations when the mode occurs at the boundary of the parameter space abstract this article gives asymptotic expansions for posterior expectations when the mode is on the boundary of the parameter space the idea based on the divergence theorem is to reduce the high dimensional integrals over the parameters space to surface integrals over the boundary of the parameter space and then apply the usual interior mode laplace method to the latter integrals it is shown that these approximations have second order accuracy the method is illustrated with applications to a two sample binomial problem and a random eflects model

a comparison of the efron hinkley ancillary and the likelihood ratio ancillary in a particular example two approximate ancillaries the efron hinkley ancillary and the signed likelihood ratio are compared in a specific example by means of their level curves their marginal densities and the conditional and unconditional likelihood functions it is shown that in the present model the latter of the two ancillaries is the better it is almost exactly distribution constant and has more stable conditional likelihood functions

second order probability matching priors for a parametric function with application to bayesian tolerance limits summary conditions are obtained for the approximate frequentist validity of the posterior quantiles of any smooth parametric function an application to bayesian tolerance limits is indicated

fixed points of commuting holomorphic mappings other than the wolff point let be the unit disc of c and let f g hol be such that f o g g o f for a 1 let fix a f p l lim r 1 f rp p lim r 1 f rp a we study the behavior of g on fix a f in particular we prove that g fix a f fix a f as a consequence besides conditions for fix a f fix a g 0 we prove a conjecture of c cowen in case f and g are univalent mappings

saddlepoint approximations for the difference of order statistics and studentized sample quantiles summary for a sample from a given distribution the difference of two order statistics and the studentized quantile are statistics whose distribution is needed to obtain tests and confidence intervals for quantiles and quantile differences this paper gives saddlepoint approximations for densities and saddlepoint approximations of the lugannani rice form for tail probabilities of these statistics the relative errors of the approximations are n i uniformly in a neighbourhood of the parameters and this uniformity is global if the densities are log concave

on robust analysis of a normal location parameter pericchi and smith considered a normal location parameter problem with double exponential and student t prior distributions these two prior distributions both belong to the class of scale mixtures of normal distributions and are useful in providing a robust analysis of the normal location parameter problem in this paper we extend the analysis to other scale mixtures of normal distributions such as the exponential power and the symmetric stable distributions

residuals in generalized linear models abstract generalized linear models are regression type models for data not normally distributed appropriately fitted by maximum likelihood rather than least squares typical examples are models for binomial or poisson data with a linear regression model for a given ordinarily nonlinear function of the expected values of the observations use of such models has become very common in recent years and there is a clear need to study the issue of appropriate residuals to be used for diagnostic purposes several definitions of residuals are possible for generalized linear models the statistical package glim baker and nelder 1978 routinely prints out residuals where v is the function relating the variance to the mean of y and is the maximum likelihood estimate of the ith mean as fitted to the regression model these residuals are the signed square roots of the contributions to the pearson goodness of fit statistic another choice of residual is the signed square root of the contribution to the devia

nd generalized map pyramids definition representations and basic operations graph pyramids are often used for representing irregular image pyramids for the 2d case combinatorial pyramids have been recently defined in order to explicitly represent more topological information than graph pyramids the main contribution of this work is the definition of pyramids of n dimensional nd generalized maps this extends the previous works to any dimension and generalizes them in order to represent any type of pyramid constructed by using any removal and or contraction operations we give basic algorithms that allow to build an nd generalized pyramid that describes a multi level segmented image a pyramid of nd generalized maps can be implemented in several ways we propose three possible representations and give conversion algorithms

on secant approximations to cumulative distribution functions summary we investigate the properties of an approximation called the secant approximation to the cumulative distribution function where the density is of a broad parametric class formulae for higher order terms are derived that give the approximation explicitly in terms of functions that define the density in all cases examined the first order secant approximation is monotonic and has the correct limits at both ends of the support resulting in the important property that the approximation itself is a cumulative distribution function we give conditions under which the approximation has bounded relative errors throughout the support the superiority of the secant approximation is illustrated in the particular case of the skew t family of distributions also known as pearson s type iv family a collection of formulae useful for the systematic evaluation of these approximations is given in appendix 2

iteration and the solution of functional equations for functions analytic in the unit disk this paper considers the classical functional equations of schroeder f o p xf and abel f o p f 1 and related problems of fractional iteration where 9p is an analytic mapping of the open unit disk into itself the main theorem states that under very general conditions there is a linear fractional transformation 4 and a function a analytic in the disk such that a oa p and that with suitable normalization 4 and a are unique in particular the hypotheses are satisfied if 4p is a probability generating function that does not have a double zero at 0 this intertwining relates solutions of functional equations for t to solutions of the corresponding equations for d for example it follows that if 4p has no fixed points in the open disk then the solution space of f o p xf is infinite dimensional for every nonzero x although the discrete semigroup of iterates of t usually cannot be embedded in a continuous semigroup of analytic functions mapping the disk into itself we find that for each z in the disk all sufficiently large fractional iterates of p can be defined at z this enables us to find a function meromorphic in the disk that deserves to be called the infinitesimal generator of the semigroup of iterates of p if the iterates of p can be embedded in a continuous semigroup we show that the semigroup must come from the corresponding semigroup for 4 and thus be real analytic in t the proof of the main theorem is not based on the well known limit technique introduced by koenigs 1884 but rather on the construction of a riemann surface on which an extension of 9p is a bijection much work is devoted to relating characteristics of p to the particular linear fractional transformation constructed in the theorem

a simple predictive density function abstract an easily computable predictive density is considered although involving a plain maximum likelihood technique it coincides with the posterior predictive density with a noninformative prior for standard situations achievements are comparable with classical inference

design and formal proof of a new optimal image segmentation program with hypermaps this article presents the design of a new functional 2d image segmentation algorithm by cell merging in a subdivision its proof of total correctness and the derivation of an optimal imperative program the planar subdivisions are modeled by hypermaps the formal specifications of hypermaps and segmentation are developed in the calculus of inductive constructions the proofs are assisted by the coq system the final program is written in c

bayesian analysis of the error correction model this paper presents a method for estimating the posterior probability density of the cointegrating rank of a multivariate error correction model a second contribution is the careful elicitation of the prior for the cointegrating vectors derived from a prior on the cointegrating space this prior obtains naturally from treating the cointegrating space as the parameter of interest in inference and overcomes problems previously encountered in bayesian cointegration analysis using this new prior and laplace approximation an estimator for the posterior probability of the rank is given the approach performs well compared with information criteria in monte carlo experiments

confidence intervals of effect size in randomized comparative parallel group studies we have presented a new likelihood based approach for constructing confidence intervals of effect size that are applicable to small samples we also conduct a simulation study to compare the coverage probability of the new likelihood based method with other three methods proposed by hedges and olkin and kraemer and paik simulation studies show that the confidence interval generated by the modified signed log likelihood ratio method possesses essentially exact coverage probabilities even for small samples although the coverage probabilities are consistently but slightly less than the nominal level the methods are also applied to two examples

estimating bayes factors via posterior simulation with the laplace metropolis estimator abstract the key quantity needed for bayesian hypothesis testing and model selection is the integrated or marginal likelihood of a model we describe a way to use posterior simulation output to estimate integrated likelihoods we describe the basic laplace metropolis estimator for models without random effects for models with random effects we introduce the compound laplace metropolis estimator we apply this estimator to data from the world fertility survey and show it to give accurate results batching of simulation output is used to assess the uncertainty involved in using the compound laplace metropolis estimator the method allows us to test for the effects of independent variables in a random effects model and also to test for the presence of the random effects

screening based on the risk of cancer calculation from bayesian hierarchical changepoint and mixture models of longitudinal markers the standard approach to early detection of disease with a quantitative marker is to set a population based fixed reference level for making further individual screening or referral decisions for many types of disease including prostate and ovarian cancer additional information is contained in the subject specific temporal behavior of the marker which exhibits a characteristic alteration early in the course of the disease in this article we derive a bayesian approach to screening based on calculation of the posterior probability of disease given longitudinal marker levels the method is motivated by a randomized ovarian cancer screening trial in the united kingdom comprising 22 000 women screened over 4 years with an additional 5 years of follow up on average levels of the antigen ca125 were recorded annually in the screened arm ca125 profiles of cases and controls from the u k trial are modeled using hierarchical changepoint and mixture models posterior distributions are calculated using markov

approximate marginal densities of nonlinear functions this paper presents an asymptotic approximation for the marginal density of a nonlinear function g 0 that is applicable when the joint density of 0 is dominated by a single mode and the jacobian of g is of full rank near that mode the approximation is based on laplace s method and its asymptotic properties are similar to those of the saddlepoint approximation the approximation is applied to the computation of a marginal posterior density a marginal sampling density and a marginal density based on a multivariate saddlepoint approximation to a joint density

posterior distributions in limited information analysis of the simultaneous equations model using the jeffreys prior abstract this paper studies the use of the jeffreys prior in bayesian analysis of the simultaneous equations model sem exact representations are obtained for the posterior density of the structural coefficient in canonical sems with two endogenous variables for the general case with m endogenous variables and an unknown covariance matrix the laplace approximation is used to derive an analytic formula for the same posterior density both the exact and the approximate formulas we derive are found to exhibit cauchy like tails analogous to comparable results in the classical literature on liml estimation moreover in the special case of a two equation just identified sem in canonical form the posterior density of is shown to have the same infinite series representation as the density of the finite sample distribution of the corresponding liml estimator this paper also examines the occurrence first documented in kleibergen and van dijk 1994a of a nonintegrable asymptotic cusp in the posterior distribution of the coefficient matrix of the reduced form equations for the included endogenous regressors an explanation for this phenomenon is provided in terms of the jacobian of the mapping from the structural model to the reduced form this interpretation assists in understanding the success of the jeffreys prior in resolving this problem

saddlepoint approximations for marginal and conditional probabilities of transformed variables the lugannani and rice formula for tail areas in the univariate case has recently been extended to tail areas for marginal distributions and for conditional distributions in certain multivariate settings however the results on relative order of errors are given only formally or under the strong continuity assumptions necessary to obtain density approximations this paper attempts to give a unified treatment of these results for smooth transformations of multivariate means under weaker conditions appropriate to indirect edgeworth approximations for probabilities

formalizing the trading theorem in coq this paper is the second half of a two part series devoted to the formalization of a combinatorial model of geometric varieties generalized maps we study here how to express continuous notions like homeomorphisms in the combinatorial world in order to constructively describe the well known theorem of surface classification according to genus number of boundaries and orientability coefficient we then use the coq specification elaborated in the previous paper to prove the first half of this theorem called the trading theorem

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

identity principles for commuting holomorphic self maps of the unit disc abstract let f g be two commuting holomorphic self maps of the unit disc if f and g agree at the common wolff point up to a certain order of derivatives no more than 3 if the wolff point is on the unit circle then f g

on bartlett adjustments for approximate bayesian inference summary in wide generality the posterior distributions of the likelihood ratio statistic and the posterior ratio statistic are chi squared to error of order o n where n is sample size the error in the chi squared approximation can be reduced to order o n 2 by bartlett correction in this paper explicit formulae are derived for the bartlett adjustment factors of both statistics and the derivations are based on the tierney kass kadane 1989 asymptotic approximation for marginal posterior probability density functions the use of numerical differentiation to facilitate calculation of the bartlett adjustments is also described some applications are considered that concern inference about regression models from both complete and right censored data

on bayesian analysis of generalized linear models using jeffreys s prior abstract generalized linear models glm s have proved suitable for modeling various kinds of data consisting of exponential family response variables with covariates bayesian analysis of such data requires specification of a prior for the regression parameters in the model used uniform priors are very commonly used as conventional noninformative priors we show however that uniform priors for glm s can lead to improper posterior distributions thus making them undesirable alternative reference priors may be constructed from jeffreys s rule in this article we give two theorems that support the use of jeffreys s priors for glm s with intrinsically fixed or known scale parameters these theorems provide i sufficient and ii necessary and sufficient conditions for the propriety of the i posterior and ii prior distributions as well as for the existence of moments implications of these theorems for some commonly used glm s are discussed finally an illustrative example is given for the binomial

some statistical properties of a family of continuous univariate distributions abstract a new two parameter family of continuous univariate distributions on the interval 1 1 is introduced and some properties are given it is shown that parameters and v are globally orthogonal in the fisher information sense and that to some extent they have the properties of a location parameter and a precision parameter respectively a pivotal statistic is constructed whose distribution is independent of the location parameter finally the connection with ultraspherical functions and brownian motion is described

a simple general formula for tail probabilities for frequentist and bayesian inference summary we describe a simple general formula for approximating the p value for testing a scalar parameter in the presence of nuisance parameters the formula covers both frequentist and bayesian contexts and does not require explicit nuisance parameterisation implementation is discussed in terms of computer algebra packages examples are given and the relationship to barndorff nielsen s approximation is discussed

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

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

frequentist and bayesian bartlett correction of test statistics based on adjusted profile likelihoods summary for inference about a multidimensional parameter b in the presence of nuisance param

analytical approximations to conditional distribution functions abstract conditional inference plays a central role in statistics but determination of relevant conditional distributions is often difficult we develop analytical procedures that are accurate and easy to apply for approximating conditional distribution functions for a continuous random vector we estimate conditional tail probabilities are smooth functions of x previous approaches have dealt with the cases where the variable whose conditional distribution is sought is a linear function of means and where there are p 1 conditioning variables however in many practical circumstances the statistic of interest is a nonlinear function of means and it is advantageous to condition on a lower dimensional ancillary statistic our procedure first involves approximating the marginal density function by an approach of phillips 1983 and tierney kass and kadane 1989 an accurate approximation to the required conditional probability is then obtained by applying a marginal tail probability approximation of diciccio and martin 1991 to the conditional density our method is illustrated in several examples including one which uses a saddlepoint approximation for the density of x and the method is applied for conditional bootstrap inference

tail probabilities from observed likelihoods summary an exponential model not in standard form is fully characterized by an observed likelihood function and its first sample space derivative up to one one transformations of the observable variable this property is used to modify the lugannani rice 1980 tail probability approximation to make it parameterization invariant then for general continuous models a version of tangent exponential model is defined and used to derive a general tail probability approximation that uses only the observed likelihood and its first sample space derivative the analysis extends from density functions to distribution functions the tangent exponential model methods of fraser 1988 a related tail probability approximation has been reported barndorff nielsen 1988b in the discussion to reid 1988

laplace expansion for posterior densities of nonlinear functions of parameters summary an asymptotic expansion for marginal posterior densities of nonlinear functions of parameters is derived the density in question is proportional to an integral over a lower dimensional manifold with the integrand including an appropriate change in volume term laplace s method is applied to obtain the expansion avoiding the difficult step of the explicit parameterization of the manifold we carry out the expansion via an implicit and local parameterization and the derivatives required can be evaluated recursively an example indicates a considerable improvement by including the term of order o n 1

approximations of marginal tail probabilities for a class of smooth functions with applications to bayesian and conditional inference summary this paper presents an asymptotic approximation of marginal tail probabilities for a real valued function of a random vector where the function has continuous gradient that does not vanish at the mode of the joint density of the random vector this approximation has error 0 n 312 and improves upon a related standard normal approximation which has error 0 n 1 derivation involves the application of a tail probability formula given by diciccio field fraser 1990 to an approximation of a marginal density derived by tierney kass kadane 1989 the approximation can be applied for bayesian and conditional inference as well as for approximating sampling distributions and the accuracy of the approximation is illustrated through several numerical examples related to such applications in the context of conditional inference we develop refinements of the standard normal approximation to the distribution of two different signed root likelihood ratio statistics for a component of the natural parameter in exponential families

polyhedra genus theorem and euler formula a hypermap formalized intuitionistic proof this article presents formalized intuitionistic proofs for the polyhedra genus theorem the euler formula and a sufficient condition of planarity they are based on a hypermap model for polyhedra and on formal specifications in the calculus of inductive constructions first a type of free maps is inductively defined from three atomic constructors next a hierarchy of types defined by invariants with operations constrained by preconditions is built on the free maps hypermaps orientated combinatorial maps and a central notion of quasi hypermaps besides the proofs of their properties are established until the genus theorem and the euler formula mainly using a simple induction principle based on the free map term algebra finally a constructive sufficient condition for polyhedra to be planar is set and proved the whole process is assisted by the interactive coq proof system

on the equivalence of prospective and retrospective likelihood methods in case control studies summary we present new approaches to analyzing case control studies using prospective likelihood methods in the classical framework we extend the equality of the profile likelihoods to the barndorff nielsen modified profile likelihoods for prospective and retrospective models this enables simple and accurate approximate conditional inference for stratified case control studies of moderate stratum size in the bayesian framework we provide sufficient conditions on priors for the prospective model parameters to yield a prospective marginal posterior density equal to its retrospective counterpart our results extend the prospective retrospective equivalence in the bayesian paradigm with a more general class of priors than has previously been investigated

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

commuting analytic functions let f and g not conformal automorphisms of the unit disk be analytic mappings of the unit disk into itself we say f and g commute if f o g g o f this paper characterizes those functions g that commute with a given function f several corollaries of this characterization give qualitative information about g given similar information about f and examples are given in each case to show the limitations of the conclusions some of the qualitative properties considered are univalence fixed point sets and whether two such g must commute with each other

the modified signed likelihood statistic and saddlepoint approximations summary for a number of tests in exponential families we show that the use of a normal approximation to the modified signed likelihood ratio statistic r is equivalent to the use of a saddlepoint approximation this is also true in a large deviation region where the signed likelihood ratio statistic r is of order in

using empirical bayes methods in biopharmaceutical research a compound sampling model where a unit specific parameter is sampled from a prior distribution and then observed are generated by a sampling distribution depending on the parameter underlies a wide variety of biopharmaceutical data for example in a multi centre clinical trial the true treatment effect varies from centre to centre observed treatment effects deviate from these true effects through sampling variation knowledge of the prior distribution allows use of bayesian analysis to compute the posterior distribution of clinic specific treatment effects frequently summarized by the posterior mean and variance more commonly with the prior not completely specified observed data can be used to estimate the prior and use it to produce the posterior distribution an empirical bayes or variance component analysis in the empirical bayes model the estimated prior mean gives the typical treatment effect and the estimated prior standard deviation indicates the heterogeneity of treatment effects in both the bayes and empirical bayes approaches estimated clinic effects are shrunken towards a common value from estimates based on single clinics this shrinkage produces more efficient estimates in addition the compound model helps structure approaches to ranking and selection provides adjustments for multiplicity allows estimation of the histogram of clinic specific effects and structures incorporation of external information this paper outlines the empirical bayes approach coverage will include development and comparison of approaches based on parametric priors for example a gaussian prior with unknown mean and variance and non parametric priors discussion of the importance of accounting for uncertainty in the estimated prior comparison of the output and interpretation of fixed and random effects approaches to estimating population values estimating histograms and identification of key considerations in the use and interpretation of empirical bayes methods

bayes inference in the tobit censored regression model we consider the bayes estimation of the tobit censored regression model with normally distributed errors a simple condition for the existence of posterior moments is provided suitable versions of monte carlo procedures based on symmetric multivariate t distributions and laplacian approximations in a certain parametrization are developed and illustrated ideas involving data augmentation and gibbs sampling cf tanner and wong 1987 and gelfand and smith 1990 are also developed the methods are compared in two examples with diffuse priors and various combinations of sample sizes and degrees of censoring

approximate interval probabilities for models of parametric dimension 1 two parameterization invariant approximations to the conditional distribution function of the maximum likelihood estimator are considered the first is derived by partial integration of a formula for the conditional density given an ancillary statistic a technique which in addition yields a useful expression for the error term the second approximation is based on an adjusted version of the signed log likelihood ratio statistic a third approximation is a modification of the first that avoids the specification of an ancillary

multiparameter testing in exponential models third order approximations from likelihood summary for composite testing of several canonical components in an exponential model a conical or directional test provides a one dimensional measure of departure we approximate the distribution of this test statistic to third order by numerical inversion of a profile likelihood the method extends the lugannani rice type formula to kernels other than the normal

assessing multivariate nonnormality using univariate distributions summary this paper proposes diagnostics that can indicate regions where multivariate distributions are poorly behaved in the sense that they are far from normal the measure of nonnormality developed by slate 1994 for univariate distributions is extended to multivariate parametric models by application to univariate marginal and conditional distributions the approach is illustrated for univariate conditional distributions using dynamic graphics in xlisp stat tierney 1990 examples show that the exploratory look at slices through the multidimensional distribution provided by the software yields considerable insight into the multivariate shape

simultaneous models for commuting holomorphic self maps of the ball abstract we prove that a finite family of commuting holomorphic self maps of the unit ball b q c q admits a simultaneous holomorphic conjugacy to a family of commuting automorphisms of a possibly lower dimensional ball and that such conjugacy satisfies a universal property as an application we describe when a hyperbolic and a parabolic holomorphic self map of b q can commute

infereni on full or partial parameters based on the standardized signed log likelihood ratio on etudie des tests et des limites de confiance bases sur la statistique du rapport de log vraisemblance signee en ajustant cette statistique de telle sorte qu a un ordre superieur d approximation la distribution normale standard soit valable

nonnormal linear regression an example of significance levels in high dimensions summary analysis of nonnormal linear models leads to an initial conditioning on the standardized residuals giving an unnormed density on rk where k is the number of parameters to obtain an observed level of significance for a single parameter it is then necessary to calculate a marginal probability thus requiring integration in k dimensions in this paper a conditional approach to evaluating the observed level of significance is developed and an importance sampling technique is used to improve the approximation and assess the accuracy of the conditional approximation to the marginal observed level a further approximation based on the invariant version fraser 1990 of the lugannani rice 1980 formula is also proposed the approach extends to the evaluation of real pivots and to bayesian inference for a single parameter component