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

approximately exact inference for the common odds ratio in several 2 2 tables abstract the conditional maximum likelihood estimator mle of the common odds ratio in a sequence of independent 2 2 tables is known to be superior to the mantel haenszel estimator in terms of asymptotic efficiency and has the further advantage that its exact distribution is known however a long standing barrier to the widespread use of this estimator has been computational intractability in particular the calculation of significance levels confidence sets and power based on the exact distribution requires fast and efficient algorithms an important class of such algorithms form the basis of statxact a software package able to solve various aspects of the exact inference problem for a sequence of several 2 2 tables in real time we provide an alternative methodology by developing several useful saddlepoint approximations to the exact distribution of the conditional mle the approximations are derived from an interesting representation for hypergeometric random variables as a sum of independent

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

practical use of higher order asymptotics for multiparameter exponential families recently developed asymptotics based on saddlepoint methods provide important practical methods for multiparameter exponential families especially in generalized linear models the aim here is to clarify and explore these attention is restricted to tests and confidence intervals regarding a single parametric function which can be represented as a natural parameter of a full rank exponential family excellent approximations to exact conditional inferences are often available in terms of simple adjustments to the signed square root of the likelihood ratio statistic

causes and explanations in the structural model approach tractable cases this paper continues the research on the computational aspects of halpern and pearl s causes and explanations in the structural model approach to this end we first explore how an instance of deciding weak cause can be reduced to an equivalent instance in which irrelevant variables in the potential weak cause and the causal model are removed which extends previous work by hopkins we then present a new characterization of weak cause for a certain class of causal models in which the causal graph over the endogenous variables has the form of a directed chain of causal subgraphs called decomposable causal graph furthermore we also identify two important subclasses in which the causal graph over the endogenous variables forms a directed tree and more generally a directed chain of layers called causal tree and layered causal graph respectively by combining the removal of irrelevant variables with this new characterization of weak cause we then obtain techniques for deciding and computing causes and explanations in the structural model approach which can be done in polynomial time under suitable restrictions this way we obtain several tractability results for causes and explanations in the structural model approach to our knowledge these are the first explicit ones they are especially useful for dealing with structure based causes and explanations in first order reasoning about actions which produces large causal models that are naturally layered through the time line and thus have the structure of layered causal graphs furthermore an important feature of the tractable cases for causal trees and layered causal graphs is that they can be recognized efficiently namely in linear time finally by extending the new characterization of weak cause we obtain similar techniques for computing the degrees of responsibility and blame and hence also novel tractability results for structure based responsibility and blame

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

approximately bayesian inference abstract consider statistical inference about a scalar parameter and suppose that information about that parameter is to be summarized by a system of interval estimates it is well known that methods of interval estimation that do not correspond to bayesian inference with respect to some prior distribution have some logical difficulties this article proposes a measure of how close a system of interval estimates is to being of bayesian form using this measure several non bayesian methods of interval estimation are analyzed these results are illustrated on several examples

saddlepoint approximation for moment generating functions of truncated random variables we consider the problem of approximating the moment generating function mgf of a truncated random variable in terms of the mgf of the underlying i e untruncated random variable the purpose of approximating the mgf is to enable the application of saddlepoint approximations to certain distributions determined by truncated random variables two important statistical applications are the following the approximation of certain multivariate cumulative distribution functions and the approximation of passage time distributions in ion channel models which incorporate time interval omission we derive two types of representation for the mgf of a truncated random variable one of these representations is obtained by exponential tilting the second type of representation which has two versions is referred to as an exponential convolution representation each representation motivates a different approximation it turns out that each of the three approximations is extremely accurate in those cases to which it is suited moreover there is a simple rule of thumb for deciding which approximation to use in a given case and if this rule is followed then our numerical and theoretical results indicate that the resulting approximation will be extremely accurate

approximations for densities of sufficient estimators abstract a simple method of obtaining asymptotic expansions for the densities of sufficient estimators is described it is an extension of the one developed by o barndorff nielsen and d r cox 1979 for exponential families a series expansion in powers of n 1 is derived of which the first term has an error of order n 1 which can effectively be reduced to n 3 2 by renormalization the results obtained are similar to those given by h e daniels s 1954 saddlepoint method but the derivations are simpler a brief treatment of approximations to conditional densities is given theorems are proved which extend the validity of the multivariate edgeworth expansion to parametric families of densities of statistics which need not be standardized sums of independent and identically distributed vectors these extensions permit the treatment of problems arising in time series analysis the technique is used by j durbin 1980 to obtain approximations to the densities of partial serial correlation coefficients

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

comparisons of approximate confidence intervals for distributions used in life data analysis this article evaluates the accuracy of approximate confidence intervals for parameters and quantiles of the smallest extreme value and normal distributions the findings also apply to the weibull and the lognormal distributions the interval estimates are based on a the asymptotic normality of the maximum likelihood estimator b the asymptotic x 2 distribution of the likelihood ratio lr statistic c a mean and variance correction to the signed square roots of the lr statistic and d the bartlett correction to the lr statistic the extensive monte carlo results about true error probabilities and average lengths under various degrees of censoring show advantages of the lr based intervals for complete or moderately censored samples the mean and variance correction to the lr statistic gives nearly exact and symmetric error probabilities in small samples with heavy censoring the bartlett correction tends to give conservative error probabilities whereas the uncorrected lr interval is often antico

general saddlepoint approximations of marginal densities and tail probabilities saddlepoint approximations of marginal densities and tail probabilities of general nonlinear statistics are derived these are based on the expansion of the statistic up to the second order their accuracy is shown in a variety of examples including logit and probit models and rank estimators for regression

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

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

approximate monte carlo conditional inference in exponential families summary this article presents an algorithm for approximate frequentist conditional inference on two or more parameters for any regression model in the generalized linear model glim family we thereby extend highly accurate inference beyond the cases of logistic regression and contingency tables implimented in commercially available software the method makes use of the double saddlepoint approximations of skovgaard 1987 journal of applied probability24 875 887 and jensen 1992 biometrika79 693 703 to the conditional cumulative distribution function of a sufficient statistic given the remaining sufficient statistics this approximation is then used in conjunction with noniterative monte carlo methods to generate a sample from a distribution that approximates the joint distribution of the sufficient statistics associated with the parameters of interest conditional on the observed values of the sufficient statistics associated with the nuisance parameters this algorithm is an alternate approach to that presented by kolassa and tanner 1994 journal of the american statistical association89 697 702 in which a markov chain is generated whose equilibrium distribution under certain regularity conditions approximates the joint distribution of interest in kolassa and tanner 1994 the gibbs sampler was used in conjunction with these univariate conditional distribution function approximations the method of this paper does not require the construction and simulation of a markov chain thus avoiding the need to develop regularity conditions under which the algorithm converges and the need for the data analyst to check convergence of the particular chain examples involving logistic and truncated poisson regression are presented

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

bayesian marginal inference abstract a method is proposed for approximating the marginal posterior density of a continuous function of several unknown parameters thus permitting inferences about any parameter of interest for nonlinear models when the sample size is finite possibly tedious numerical integrations are replaced by conditional maximizations which are shown to be quite accurate in a number of special cases there are similarities with the profile likelihood ideas originated by kalbfleisch and sprott 1970 and the method is contrasted with a laplacian approximation recommended by kass tierney and kadane 1988 in press referred to here as the ktk procedure the methods are used to approximate the marginal posterior densities of the log linear interaction effects and an overall measure of association in a two way contingency table snee s 1974 hair eye color data are reanalyzed and adjustments are proposed to goodman s 1964 analysis for the full rank interaction model another application concerns marginaliz

approximate conditional inference in generalized linear models on developpe des approximations precises et facilement calculables pour les densites conditionnelles et les distributions des statistiques exhaustives dans les modeles lineaires generalises avec des fonctions de liaison canoniques

a framework for bayesian and likelihood approximations in statistics summary regular bayesian and frequentist approximations in statistics are studied within a unified framework in particular it is shown how some higher order likelihood based approximations arise from their bayesian counterparts via an unsmoothing argument this approach serves to shed new light on these formulae and to clarify relationships between bayesian and frequentist inferences for example bayesian interpretations of higher order approximations can give insights into the acceptability or otherwise of these approximations from the point of view of relevance to the actual data observed furthermore this approach is a very natural one for the study of more general nonregular problems models for dependent data and approximate conditional inference for ease of exposition the development is in terms of a single real parameter the main analytic development proceeds in terms of signed roots of log density ratios

fast and accurate approximate double bootstrap confidence intervals we propose two new methods for the construction of approximate iterated bootstrap confidence intervals our methods follow a saddlepoint approach but employ approximate rather than exact solutions to the nonlinear saddlepoint equations that arise in applying that technique the main advantages of the new techniques are that they yield confidence intervals with high coverage accuracy in small to moderately sized samples and that they achieve this accuracy with very substantial reductions in computation time compared to other methods

saddlepoint approximation for the distribution of a ratio of quadratic forms in normal variables abstract in this article the saddlepoint approximations to the density and tail probability of a ratio of quadratic forms in normal variables are derived a numerical exposition via the durbin watson test statistic reveals several desirable features the approximations which involve only a limited number of computable functions provide the practitioner with an accessible and a very powerful tool

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

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

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

on the corrections to scaling in three dimensional ising models the leading correction to scaling amplitudes for the spin 1 2 nearest neighbor sc bcc and fee ising models are considered with the particular aim of determining their signs on the basis of previous two variable series analyses by chen fisher and nickel and renormalization group 4 d expansions it is concluded that the correction amplitudes for the susceptibility correlation length specific heat and spontaneous magnetization arenegative for all three lattices thus for example the effective exponent eff t asymptotically approaches the true susceptibility exponent fromabove other earlier and more recent high temperature series and field theoretic analyses are seen to be consistent with this result however the usual nonasymptotic perturbative field theoretic approaches are essentially committed to positive correction amplitudes the question of the signs therefore relates directly to the applicability of these non asymptotic field theoretic calculations to three dimensional ising models as well as to different experimental systems

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

a bayesian approach to approximate conditional inference summary sweeting 1995 studies regular bayesian and frequentist approximations within a unified framework in the case of a single parameter and shows that higher order approximations to sampling distributions arise from their bayesian counterparts via an unsmoothing argument in the present paper we extend this programme to include formulae in approximate conditional inference in particular it is shown how bayesian arguments may be used to derive some formulae developed by barndorff nielsen 1980 1983 1986 the development proceeds in terms of likelihood roots

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

default priors for bayesian and frequentist inference summary we investigate the choice of default priors for use with likelihood for bayesian and frequentist inference such a prior is a density or relative density that weights an observed likelihood function leading to the elimination of parameters that are not of interest and then a density type assessment for a parameter of interest for independent responses from a continuous model we develop a prior for the full parameter that is closely linked to the original bayes approach and provides an extension of the right invariant measure to general contexts we then develop a modified prior that is targeted on a component parameter of interest and by targeting avoids the marginalization paradoxes of dawid and co workers this modifies jeffreys s prior and provides extensions to the development of welch and peers these two approaches are combined to explore priors for a vector parameter of interest in the presence of a vector nuisance parameter examples are given to illustrate the computation of the priors