lagrangian hamiltonian unified formalism for field theory the rusk skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems it incorporates all the characteristics of lagrangian and hamiltonian descriptions of these systems including dynamical equations and solutions constraints legendre map evolution operators equivalence etc in this work we extend this unified framework to first order classical field theories and show how this description comprises the main features of the lagrangian and hamiltonian formalisms both for the regular and singular cases this formulation is a first step toward further applications in optimal control theory for partial differential equations

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

bayesian and frequentist bartlett corrections for likelihood ratio and conditional likelihood ratio tests summary this paper characterizes priors under which the bayesian and frequentist bartlett corrections for the likelihood ratio and the conditional likelihood ratio clr statistics differ by o 1 it is seen that except for sample points with negligible probability the clr statistic has a posterior distribution for which a posterior bartlett correction exists this observation leads to an alternative proof of the existence of a frequentist bartlett correction for the clr statistic

bayesian inference for small sample capture recapture data we consider data on the survival of a population of cephalorhynchus hectori hector s dolphins in a marine area of new zealand to estimate survival probabilities of animal populations a multiple capture recapture sampling scheme can be used in this paper we propose a practical methodology to derive approximations to posterior distributions based on laplace methods we show how to calculate bayes estimates and credible intervals in this setting

hamilton jacobi theory in multisymplectic classical field theories the geometric framework for the hamilton jacobi theory developed in the studies of carinena et al int j geom methods mod phys 3 7 1417 1458 2006 carinena et al int j geom methods mod phys 13 2 1650017 2015 and de leon et al variations geometry and physics nova science publishers new york 2009 is extended for multisymplectic first order classical field theories the hamilton jacobi problem is stated for the lagrangian and the hamiltonian formalisms of these theories as a particular case of a more general problem and the classical hamilton jacobi equation for field theories is recovered from this geometrical setting particular and complete solutions to these problems are defined and characterized in several equivalent ways in both formalisms and the equivalence between them is proved the use of distributions in jet bundles that represent the solutions to the field equations is the fundamental tool in this formulation some examples are analyzed and in particular the h

on the insensitivity of the score test for heterogeneity to omitted covariates in multivariate failure time models the multivariate weibull distribution often provides a point of departure for modelling repeated or a sequence of failure times for a particular individual it is then natural to want to test the implied assumption of homogeneity and a number of authors have derived a score test for this purpose see for example kimber 1996 in the present paper it is shown that the proposed score test for neglected heterogeneity can be insensitive to specification error in the implied log linear regression structure

approximate bayesian analysis of censored survival data summary the bayesian analysis of data arising from a location scale regression model and which may include a number of right censored observations is considered the posterior distributions of the suitably normalized precision parameter and regression coefficient vector are shown to be well approximated by x2 and multivariate t distributions respectively the ideas are illustrated using some clinical survival data

saddlepoint expansions for directional test probabilities an expansion is derived for the conditional tail probability of a multivariate random variable given its direction from the expected value in particular if the score function of a statistical model is chosen as this variable such a conditional probability gives a test for a simple hypothesis which is asymptotically equivalent to the likelihood ratio test the expansion is of the large deviation type and is derived through the use of saddlepoint methods thus the relative error of the test probability is o n1 uniformly in a bounded set for an average of n independent replications the approximation is based on the cumulant transform for the random variable but is otherwise a simple expression in terms of chi squared distributions in the one dimensional case it reduces to the expansion obtained by lugannani and rice numerical examples show an excellent fit comparable with other saddlepoint expansions when the dimension is low even for very small sample sizes but for higher dimensions more replications are required to give a similar approximation

on the legendre map in higher order field theories the authors show how the construction of a cartan form in higher order field theories defines a legendre map and how the regularity of this map may be described in terms of a sequence of maps between intermediate phase spaces they also show how semi holonomic jets are important in this context they allow the definition of regularity to be applied without ambiguity to the lagrangian itself and they permit the specification of a unique cartan form legendre map and covariant phase space for second order lagrangians

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

estimating functions and approximate conditional likelihood summary the approximate conditional likelihood method proposed by cox reid 1987 is applied to the estimation of a scalar parameter 0 in the presence of nuisance parameters the estimating function of 0 based on the approximate conditional likelihood is shown to be preferable to that based on the profile likelihood a sufficient condition for both approaches to be equivalent is given the role of parameter orthogonality is emphasized several examples including bivariate normal means with known coefficient of variation are presented

the analysis of transformed data abstract recently it was suggested bickel and doksum 1981 that when data are used to select a transformation the post transformation analysis of those data may need to be modified considerably from standard form so as to allow for the selection we argue that common sense and the work of box and cox 1964 point to a contrary conclusion our argument is based on considerations of parameter interpretation and subsequent bayesian analysis within the context of fitting normal error linear models numerical examples are used to illustrate the main points

an analysis of transformations revisited rebutted abstract in a paper written in 1964 box and cox described a method for estimating transformations and showed how in suitable cases valuable increases in simplicity and efficiency were possible since that time this technique has enjoyed wide practical use and considerable success however a recent theoretical paper by bickel and doksum 1981 seems to suggest that serious dangers are associated with the employment of this method and speaks of instability and cost of estimation of the transformation these difficulties seem to be associated with 1 examples which common sense would rule out namely situations where the effect of transformation on the data is almost linear so that it is a matter of indifference which transformation is used 2 the idea that it makes sense to state conclusions in terms of a number measured on an arbitrary scale 3 failure to take proper account of the jacobian of the transformation

interval estimation based on the profile likelihood strong lagrangian theory with applications to discrimination summary strong lagrangian theory is used to illuminate the properties of the profile log likelihood general conditions are given which lead to a simplification of the computations required to plot this function or calculate interval estimates based upon it the results presented are rather general with a variety of possible applications in particular it is shown that they provide a simple solution to the important practical problem of obtaining an interval estimate for the posterior log odds ratio in the two population discrimination problem the results are applied to the multivariate normal unequal covariance matrix case and are illustrated by an application

generalized hamiltonian dynamics i formulation on t q tq the dirac bergmann generalized hamiltonian dynamics for a degenerate lagrangian system is formulated on the whitney sum t q tq of the phase space t q and the velocity space tq over the configuration space q the formulation is related to those on t q and tq some ambiguities concerning generalized dynamics that have appeared in the literature are clarified

on the choice of prior distribution for the box cox transformed linear model summary the noninformative prior distribution for the parameters of the linear model transformed following box cox 1964 has the non bayesian property of depending to some extent on the data an alternative choice of prior which is not outcome dependent was suggested by pericchi 1981 but it is argued here that this prior has some undesirable features an alternative family of non outcome dependent priors is suggested leading to a noninformative prior which is closer in spirit to that proposed by box cox the posterior consequences of adopting this prior are fully explored and an example discussed

nonparametric functional mapping of quantitative trait loci functional mapping is a useful tool for mapping quantitative trait loci qtl that control dynamic traits it incorporates mathematical aspects of biological processes into the mixture model based likelihood setting for qtl mapping thus increasing the power of qtl detection and the precision of parameter estimation however in many situations there is no obvious functional form and in such cases this strategy will not be optimal here we propose to use nonparametric function estimation typically implemented with b splines to estimate the underlying functional form of phenotypic trajectories and then construct a nonparametric test to find evidence of existing qtl using the representation of a nonparametric regression as a mixed model the final test statistic is a likelihood ratio test we consider two types of genetic maps dense maps and general maps and the power of nonparametric functional mapping is investigated through simulation studies and demonstrated by examples

reductions of topologically massive gravity i hamiltonian analysis of second order degenerate lagrangians we present skinner rusk and hamiltonian formalisms of second order degenerate clement and sarioglu tekin lagrangians the dirac bergmann constraint algorithm is employed to obtain hamiltonian realizations of lagrangian theories the gotay nester hinds algorithm is used to investigate skinner rusk formalisms of these systems

on the extraction of the boundary conditions and the boundary ports in second order field theories in this paper we consider second order field theories in a variational setting from the variational principle the euler lagrange equations follow in an unambiguous way but it is well known that this is not true for the cartan form this also has consequences on the derivation of the boundary conditions when non trivial variations are allowed on the boundary by posing extra conditions on the set of possible boundary terms we exploit the degree of freedom in the cartan form to extract physical meaningful boundary expressions the same mathematical machinery will be applied to derive the boundary ports in a hamiltonian representation of the partial differential equations which is crucial for energy based control approaches our results will be visualized for mechanical systems such as beam and plate models in this paper we consider second order field theories in a variational setting from the variational principle the euler lagrange equations follow in an unambiguous way but it is well known that this is not true for the cartan form this also has consequences on the derivation of the boundary conditions when non trivial variations are allowed on the boundary by posing extra conditions on the set of possible boundary terms we exploit the degree of freedom in the cartan form to extract physical meaningful boundary expressions the same mathematical machinery will be applied to derive the boundary ports in a hamiltonian representation of the partial differential equations which is crucial for energy based control approaches our results will be visualized for mechanical systems such as beam and plate models

properties of sufficiency and statistical tests 1 in a previous paper dealing with the importance of properties of sufficiency in the statistical theory of small samples attention was mainly confined to the theory of estimation in the present paper the structure of small sample tests whether these are related to problems of estimation and fiducial distributions or are of the nature of tests of goodness of fit is considered further

maximum likelihood estimation and inference for high dimensional generalized factor models with application to factor augmented regressions abstract this paper reestablishes the main results in bai 2003 and bai and ng 2006 for generalized factor models with slightly stronger conditions on the relative magnitude of n number of subjects and t number of time periods convergence rates of the estimated factor space and loading space and asymptotic normality of the estimated factors and loadings are established under mild conditions that allow for linear logit probit tobit poisson and some other single index nonlinear models the probability density mass function is allowed to vary across subjects and time thus mixed models are also allowed for for factor augmented regressions this paper establishes the limit distributions of the parameter estimates the conditional mean and the forecast when factors estimated from nonlinear mixed data are used as proxies for the true factors

bias reduction in nonlinear and dynamic panels in the presence of cross section dependence abstract fixed effects estimation of nonlinear dynamic panel models is subject to the incidental parameter issue leading to a biased asymptotic distribution while this problem has been studied extensively in the literature a general analysis allowing for both serial and cross sectional dependence is missing in this paper we investigate the large n t theory of the profile and integrated likelihood estimators allowing for dependence across both dimensions we show that under stronger dependence types the asymptotic bias disappears but a o p 1 t small sample bias remains we provide bias correction and inference methods and also obtain primitive conditions for asymptotic normality under various dependence settings

a simple method for the adjustment of profile likelihoods summary we propose a simple adjustment for profile likelihoods the aim of the adjustment is to alleviate some of the problems inherent in the use of profile likelihoods such as bias inconsistency and overoptimistic variance estimates the adjustment is applied to the profile loglikelihood score function at each parameter value so that its mean is zero and its variance is the negative expected derivative matrix of the adjusted score function for cases in which explicit calculation of the adjustments is difficult we give two methods to simplify their computation an automatic simulation method that requires as input only the profile loglikelihood and its first few derivatives first order asymptotic expressions some examples are provided and a comparison is made with the conditional profile log likelihood of cox and reid

nonhomogeneity analysis using borrowed strength abstract this article develops a borrowed strength methodology for estimating local probability densities in a random field under a piecewise stationarity condition similarities between the densities in different regions of the field can be exploited by estimating a global mixture density and imposing the parameter space support of this borrowed strength estimate on the local estimation problems the local estimates are then used in an analysis of the homogeneity of the field which is shown to benefit from the method of borrowing strength

bartlett type adjustment for the conditional likelihood ratio statistic of cox and reid summary this paper explicitly derives a bartlett type adjustment for the conditional likelihood ratio statistic of cox reid via that for the usual likelihood ratio statistic

reductions of topologically massive gravity ii first order realizations of second order lagrangians second order degenerate clement and sarioglu tekin lagrangians are casted into forms of various first order lagrangians the structures of the iterated tangent bundle and acceleration bundle are presented as a suitable geometric framework hamiltonian analyses of these equivalent formalisms are performed by means of the dirac bergmann constraint algorithm all formulations are shown to possess only second class constraints

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

constrained variational calculus for higher order classical field theories we develop an intrinsic geometrical setting for higher order constrained field theories as a main tool we use an appropriate generalization of the classical skinner rusk formalism some examples of applications are studied in particular to the geometrical description of optimal control theory for partial differential equations

predictive likelihood inference with applications summary conditional and marginal predictive likelihood are defined and used in predicting un observed or missing data a distinction is made between predictive and estimative goals of a data analysis such methodology and distinctions are illustrated in the context of outlier theory and mixed model ano va the prediction of unobserved or missing data is considered from a likelihood perspective a general predictive likelihood is proposed for such purposes and contrasted with parametric likelihood the resulting maximum likelihood predictor is also compared with the limiting imputation from the em algorithm the new methodology is illustrated and contrasted with estimative likelihood analysis in the context of outlier theory and mixed model ano va predictions are proposed that are based on conditional and marginal predictive likelihoods for the unobserved data which are predictive adaptations of the methods developed by kalbfleisch and sprott 1970 1973 these authors were concerned with the elimination of nuisance parameters in order to make inference about another set of structural parameters prediction in the frame work of parametric distributions involves the elimination of all parameters in such a way that inference about the unobserved data is possible therefore in the predictive setting all para meters are nuisance parameters and the unobserved data now play a role similar to that of the structural parameters various predictive approaches to statistical inference have been advocated principally by geisser 1971 1975 stone 1974 aitchison and dunsmore 1975 and guttman 1970 a survey is provided by geisser 1980 and a brief overview in a parametric setting is presented in the next section following this a conditional likelihood approach is developed and shown to be closely related to the lauritzen 1974 hinkley 1979 likelihood method but specified in a manner allowing for a wider range of applicability a marginal predictive likelihood procedure is also specified which when applicable essentially agrees with the conditional likelihood procedure for the examples considered for a general account of likelihood based inference see fraser

first order equations of motion for classical mechanics a global canonical first order equation of motion is derived for any mechanical system obeying newton s second law the existence of a lagrangian is not assumed but the properties of the canonical equation are similar to those of the hamiltonian formulation the choice of map f from velocity space to phase space is not determined by the condition that the first order equation of motion be equivalent to a second order equation on configuration space and therefore is left open to be selected on the basis of other considerations the canonical equation is a covector or 1 form equation on the whitney sum t q tq and contains the second order equation condition restriction to the graph of the map f and newton s equation of motion in first order form the last is related to newton s second order equations by the consistency condition that the motions not lead off graph f in t q tq the first order equation of motion can be projected onto phase space if the map f can be inverted

a second order adjustment to the profile likelihood in the case of a multidimensional parameter of interest summary inference in the presence of nuisance parameters is often carried out by using the x2approximation to the profile likelihood ratio statistic however in small samples the accuracy of such procedures may be poor in part because the profile likelihood does not behave as a true likelihood in particular having a profile score bias and information bias which do not vanish to account better for nuisance parameters various researchers have suggested that inference be based on an additively adjusted version of the profile likelihood function each of these adjustments to the profile likelihood generally has the effect of reducing the bias of the associated profile score statistic however these adjustments are not applicable outside the specific parametric framework for which they were developed in particular it is often difficult or even impossible to apply them where the parameter about which inference is desired is multidimensional in this paper we propose a new adjustment function which leads to an adjusted profile likelihood having reduced score and information biases and is readily applicable to a general parametric framework including the case of vector valued parameters of interest examples are given to examine the performance of the new adjusted profile likelihood in small samples and also to compare its performance with other adjusted profile likelihoods

approximate predictive likelihood summary a predictive likelihood is given which approximates both bayes and maximum likeli hood predictive inference by expansion of a posterior likelihood this synthesizes and extends previous results and is widely applicable the approximation usually differs from exact bayes posterior predictive density by op n 2 and from exact predictive likelihood by op n 1 but does not depend on the availability of prior information and is applicable when exact predictive likelihood cannot be found the results are applied to the prediction various treatments of the likelihood function form the basis of parametric inference but until recently few non bayesian attempts have been made to define versions of it suitable for prediction however lauritzen 1974 hinkley 1979 and more recently butler 1986 have proposed rather similar definitions of predictive likelihood based on the idea of conditioning on the value of a sufficient statistic unfortunately they apply only when sufficiency provides a genuine reduction of the data moreover even if for a given problem an exact predictive likelihood exists the calculations needed to derive it can be onerous this paper provides approximations to hinkley lauritzen predictive likelihood accur ate to op n 1 in many cases which are applicable in situations where maximum likelihood estimation is regular they also apply to problems which admit no exact predictive likelihood because sufficiency provides no real reduction of the data the approximations are derived from a posterior predictive density and may if prior informa tion is available be regarded as bayesian procedures their construction requires only the repeated maximization of a likelihood and evaluation of the observed information matrix at the maximum and is fast and accurate using usual numerical maximization methods there is a close connexion with the approximations to posterior moments and marginal densities given by tierney kadane 1986 who suggest the same approxima tion as used here but give no detailed results for prediction section 2 of this paper shows how laplace s method for integrals may be applied to a posterior predictive density to yield approximate predictive likelihood section 3 discusses a connection with conditional inference essentially the same approximation

consistent estimation of limited dependent variable models despite misspecification of distribution abstract by exploiting conditions on the behavior of regressors estimators are developed which are consistent up to a factor of proportionality for the slope coefficients of the regression function of a imperfectly observed dependent variable limited dependent variables are a special case of interest the consistency is robust to misspecification of the distribution of the latent dependent variable the estimators are weighted pseudo maximum likelihood estimators which can be readily computed using popular methods and software

practical small sample asymptotics for regression problems abstract saddlepoint approximations are derived for sums of independent but not necessarily identically distributed random variables along with generalizations to estimating equations and multivariate problems these results are particularly useful for accurately approximating the distribution of regression coefficients general formulas are given for the distribution of the coefficients arising out of a generalized linear model with both canonical and noncanonical link functions we illustrate the case of logistic regression with a real data example and show how the gibbs sampler may be used to obtain confidence sets for each regression parameter based on the saddlepoint approximation

a note on the relation between modified profile likelihood and the cox reid adjusted profile likelihood summary an adjustment to the profile likelihood proposed by cox reid 1987 when the parameters are orthogonal is shown to agree with modified profile likelihood in a number of instances in which the parameters are not orthogonal several examples of the phenomenon are given in the case of composite transformation models with invariant parameter as the parameter of interest it is shown that the cox reid adjustment agrees with modified profile likelihood in a large deviation sense if lebesgue measure is rightinvariant necessary and sufficient conditions for equivalence in the moderate deviation sense of modified profile likelihood and the cox reid adjusted profile likelihood are

assessment of local influence summary statistical models usually involve some degree of approximation and therefore are nearly always wrong because of this inexactness an assessment of the influence of minor perturbations of the model is important we discuss a method for carrying out such an assessment the method is not restricted to linear regression models and it seems to provide a relatively simple unified approach for handling a variety of problems

adjusted versus conditional likelihood power properties and bartlett type adjustment summary under contiguous alternatives the adjusted likelihood ratio alr test is seen to have the same power properties as the conditional likelihood ratio test up to the third order of comparison in particular an optimum property of the alr test in terms of second order local maximinity follows it is also seen that the alr statistic admits a bartlett type adjustment

approximate inference in location scale regression models abstract it is shown that 2 and multivariate t distributions provide good approximations to posterior distributions arising in location scale regression models in the case of a linear regression model bayesian inferences based on an improper prior have conditional frequency as well as fiducial structural interpretation so the results are also applicable in a non bayesian context numerical support for the approximations is presented for the case of a weibull regression model and a regression with t distributed errors

conditional score functions some optimality results summary the conditional score function has previously been shown to generate the optimal estimating equation for a parameter of interest when the conditioning statistic is complete and sufficient for the nuisance parameters godambe 1976 the present paper generalizes these results to partial likelihood factorizations and then examines the nature of the problem when the appropriate conditioning statistics depend on the parameter of interest in this case globally optimal estimating functions are impossible a weaker criterion of optimal weighting leads to a class of estimated conditional score functions which satisfy an information equality the purpose of this paper is to develop and justify an approach based on conditional likelihood to estimation in the presence of nuisance parameters the arguments focus on the case when there is a complete and sufficient statistic s 0 for the nuisance parameters when the parameter of interest 0 is fixed an extended notion of completeness will be presented in the context of partial likelihoods the problem takes on two separate natures depending on whether the statistic s 0 truly depends on 0 or not if s s 0 is free of 0 then the conditional likelihood of the data given s has been shown to generate optimal inferential procedures giving a best estimating equation godambe 1976 and a best asymptotically normal estimator in a mixture setting lindsay 1980 if s 0 depends on 0 then it will be shown that the best estimating equation also a conditional likelihood score will generally be only locally optimal leaving no single best strategy consideration of an optimal weighting strategy yields estimators which have second order properties similar to the conditional likelihoods of the first type

the score statistic for regression transformation summary the score statistic for regression transformation using a parametric family is developedfurther a simple formula for the approximate variance matrix of score statistics basedon asymptotic theory is applied to yield the score statistic t s for regression transforma tion this is compared to atkinson s t d statistic for the same problem and yields animprovement to its standardization simulations indicate that the significance levels ofthe proposed score statistic t s are very close to their nominal values some key words power transformation regression score statistic simulation tests for normality 1 introduction the work of box cox 1964 has been the basis of many investigations into theuse of transformations in regression analysis and the current state of the topic is wellsummarized in the monographs of cook weisberg 1982 and atkinson 1985 theprincipal aim of the original work was to transform a response or dependent variable sothat it would satisfy the normality and other standard assumptions of regression analysis the authors used a parametric family of transformations to achieve this and placedemphasis on estimating its parameters in subsequent work andrews 1971 was moreconcerned with testing transformation parameters while atkinson 1973 introducedanother statistic for this purpose from consideration of both andrews s statistic and scoreor c a tests the present paper develops a standardized score statistic in the case of ascalar transformation parameter in effect it is found to provide the standardization ofatkinson s statistic an aspect which atkinson 1973 believed to be theoretically intract able and avoided by using an approximate regression argument

conditional likelihood and power higher order asymptotics this paper considers in the presence of a nuisance parameter a very large class of tests that includes the conditional and the usual versions of the likelihood ratio lr rao s and wald s tests under contiguous alternatives and orthogonal parametrization the power functions of the conditional and the usual versions of these tests have been compared and in particular it is seen that the power functions of the conditional versions unlike those of the usual versions are identical up to the second order with the power functions of the corresponding tests with known nuisance parameter an optimality property of the conditional lr test in terms of second order local maximinity has been established a test optimal in the sense of third order average power under contiguous alternatives has been proposed a weaker optimality property of rao s test in terms of third order average power has also been indicated

scale parameters a bayesian treatment a definition of a scale parameter is given which includes the usual definition of a scale parameter in the exponential family some suggestions for the estimation of and approximations to the marginal posterior distribution of a vector valued parameter of primary interest are made in particular the possibility of a 2 approximation to the posterior distribution of leading to h p d regions for based on the f distribution is explored two examples are discussed where observations are from a normal and gamma distribution

bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator for rather general parametric models a simple connection is established between the bartlett adjustment factor of the log likelihood ratio statistic and the normalizing constant c of the formula c i i 1 2l for the conditional distribution of a maximum likelihood estimator as applied to the full model and the model of the hypothesis tested this leads to a relatively simple demonstration that division of the likelihood ratio statistic by a suitable constant or estimated factor improves the chi squared approximation to its distribution various expressions for these quantities are discussed in particular for the case of a one dimensional parameter an approximation to the constants involved is derived which does not require integration over the sample space

primes in arithmetic progressions strengthening work of rosser schoenfeld and mccurley we establish explicit chebyshev type estimates in the prime number theorem for arithmetic progressions for all moduli k 72 and other small moduli

effect of misspecification of regression models in the analysis of survival data the accelerated life and proportional hazards families of models are widely used in the analysis of survival data we examine the question if interest lies in the qualitative effect on failure of various explanatory variables how critical is the choice of model family in assessing the relative importance of the explanatory variables we consider small effects when censoring is independent of the explanatory variables to first order the regression parameters are proportional under the alternative models this is not true when the censoring varies appreciably with the explanatory variables some higherorder theory is examined for the special case of the two sample problem

a bayesian approach to transformations to normality summary the analysis of transformation of observations in the linear model with normal errors proposed by box cox 1964 is considered a different choice of noninformative unnormed prior is advocated which is not outcome dependent this new selection of prior leads to a formal identity between likelihood and bayesian inference both for the estimation of the best transformation to normality and for the presence of homoscedasticity and additivity under this transformation extension to a related problem is mentioned

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

differential geometry of curved exponential families curvatures and information loss the differential geometrical framework is given for analyzing statistical problems related to multi parameter families of distributions the dualistic structures of the exponential families and curved exponential families are elucidated from the geometrical viewpoint the duality connected by the legendre transformation is thus extended to include two kinds of affine connections and two kinds of curvatures the second order information loss is calculated for fisher efficient estimators and is decomposed into the sum of two non negative terms one is related to the exponential curvature of the statistical model and the other is related to the mixture curvature of the estimator only the latter term depends on the estimator and vanishes for the maximum likelihood estimator a set of statistics which recover the second order information loss are given the second order efficiency also is obtained the differential geometry of the function space of distributions is discussed

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

some remarks on the abc conjecture let r x be the product of all distinct primes dividing a nonzero integer x the abc conjecture says that if a b c are nonzero relatively prime integers such that a b c 0 then the biggest limit point of the numbers logmax lal ibl cil log r abc equals 1 we show that in a natural anologue of this conjecture for n 3 integers the largest limit point should be replaced by at least 2n 5 we present an algorithm leading to numerous examples of triples a b c for which the above quotients strongly deviate from the conjectural value 1

using hierarchical likelihood for missing data problems most statistical solutions to the problem of statistical inference with missing data involve integration or expectation this can be done in many ways directly or indirectly analytically or numerically deterministically or stochastically missing data problems can be formulated in terms of latent random variables so that hierarchical likelihood methods of lee nelder 1996 can be applied to missing value problems to provide one solution to the problem of integration of the likelihood the resulting methods effectively use a laplace approximation to the marginal likelihood with an additional adjustment to the measures of precision to accommodate the estimation of the fixed effects parameters we first consider missing at random cases where problems are simpler to handle because the integration does not need to involve the missing value mechanism and then consider missing not at random cases we also study tobit regression and refit the missing not at random selection model to the antidepressant trial data analyzed in diggle kenward 1994

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

likelihood and observed geometries a partir du point de vue de la conditionnalite on etablit un parallele observe a la theorie de chentsov efron amari etc en utilisant l information observee et un tenseur d asymetrie observee

bayesian intervals with good frequentist behaviour in the presence of nuisance parameters given a random sample from a distribution with density function that depends on an unknown parameter 1 p we are concerned with the problem of setting confidence intervals for a particular component of it say 1 treating the remaining components as a nuisance parameter adopting an objective bayesian approach we show that the bayes intervals with a certain conditional prior density on the parameter of interest 1 are confidence intervals as well having nearly the correct frequency of coverage the frequentist performance of the proposed intervals is tested in a simulation study for gamma mean and shape parameters

higher order lagrangian systems geometric structures dynamics and constraints in order to study the connections between lagrangian and hamiltonian formalisms constructed from a perhaps singular higher order lagrangian some geometric structures are constructed intermediate spaces between those of lagrangian and hamiltonian formalisms partial ostrogradski s transformations and unambiguous evolution operators connecting these spaces are intrinsically defined and some of their properties studied equations of motion constraints and arbitrary functions of lagrangian and hamiltonian formalisms are thoroughly studied in particular all the lagrangian constraints are obtained from the hamiltonian ones once the gauge transformations are taken into account the true number of degrees of freedom is obtained both in the lagrangian and hamiltonian formalisms and also in all the intermediate formalisms herein defined

unambiguous formalism for higher order lagrangian field theories the aim of this paper is to propose an unambiguous intrinsic formalism for higher order field theories which avoids the arbitrariness in the generalization of the conventional description of field theories and implies the existence of different cartan forms and legendre transformations we propose a differential geometric setting for the dynamics of a higher order field theory based on the skinner and rusk formalism for mechanics this approach incorporates aspects of both the lagrangian and the hamiltonian description since the field equations are formulated using the lagrangian on a higher order jet bundle and the canonical multisymplectic form on its affine dual as both of these objects are uniquely defined the skinner rusk approach has the advantage that it does not suffer from the arbitrariness in conventional descriptions the result is that we obtain a unique and global intrinsic version of the euler lagrange equations for higher order field theories several examples illustrate our construction

simultaneous rational approximation to binomial functions we apply pade approximation techniques to deduce lower bounds for simultaneous rational approximation to one or more algebraic numbers in particular we strengthen work of osgood fel dman and rickert proving for example that max 2 p1 q 3 p2 q q 1 79155 for q q0 where the latter is an effective constant some of the diophantine consequences of such bounds will be discussed specifically in the direction of solving simultaneous pell s equations and norm form equations 0 introduction in 1964 baker 1 2 utilized the method of pade approximation to hypergeometric functions to obtain explicit improvements upon liouville s theorem on rational approximation to algebraic numbers by way of example he showed that 3 2 pq 10 6q 2 955 0 1 for all positive integers p and q and used such bounds to solve related diophantine equations chudnovsky 6 subsequently refined baker s results primarily through a detailed analysis of the arithmetical properties of certain pade approximants analogous to 0 1 he proved that 3 2 pq q 2 42971 0 2 for all integers p and q with q greater than some effectively computable constant q0 by working out the implicit constants in 0 2 easton 8 deduced 3 2 pq 6 6 10 6q 2 795 for positive integers p and q as well as related bounds for other cubic irrationalities similar results exist for simultaneous approximation to several algebraic numbers in particular baker 3 derived bounds of the form max 1 u m u pu q q 0 3 received by the editors june 30 1994 and in revised form january 31 1995 1991 mathematics subject classification primary 11j68 11j82 secondary 11d57

an approximation to maximum likelihood estimates in reduced models summary an approximation to the maximum likelihood estimates of the parameters in a model can be obtained from the corresponding estimates and information matrices in an extended model i e a model with additional parameters the approximation is close provided that the data are consistent with the first model applications are described to log linear models for discrete data to models for multivariate normal distributions with special covariance matrices and to mixed discrete continuous models

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

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

parametric estimation procedures for screening programmes stable and nonstable disease models for multimodality case finding summary this paper develops methods for estimating the parameters associated with early detection programmes disease is considered to have three states a disease free state or a state in which the disease cannot be detected a preclinical state and a clinical state the natural history of the disease is assumed to be progressive the parameters to be estimated are the sensitivity of one or two disease detection modalities and the characteristics of the preclinical sojourn time distribution under both the stable disease and nonstable disease models the stable disease model assumes that the incidence or prevalence of a disease is independent of age or chronological time while the nonstable disease model allows these quantities to depend on time with the nonstable disease model the relevant parameters can be jointly estimated by a two step iteration procedure from the likelihood function for the stable disease model the sensitivity and the parameters of the sojourn time distribution of the preclinical state can be obtained directly from a conditional likelihood function applications are made to recent clinical trials for the early detection of breast cancer

effective irrationality measure for certain algebraic numbers a result of chudnovsky concerning rational approximation to certain algebraic numbers is reworked to provide a quantitative result in which all constants are explicitly given more particularly pade approximants to the function 1 x 113 are employed to show for certain integers a and b that j a b 1 3 p q cq k when q 0 here c and k are given as functions of a and b only in 1964 baker 1 improving a technique used by siegel 8 was able to obtain effective irrationality measures for the function 1 x m n evaluated at certain rational points in particular he was able to show that for integers p q we have 1 121 3 p q 10 6q 2955 when q 0 the technique was further refined by chudnovsky 2 whose results when applied to 21 3 imply that for any 0 there exists a positive integer qo c such that for integers p q we have 2 121 3 p qi q 2 429 e when q q0 chudnovsky s result is effective in the sense that it is possible in principle to work through the proof and compute for any particular value of a qo c for which 2 holds however chudnovsky does not undertake such computations in this article we rework baker s proof using chudnovsky s refinement together with a chebyshev type result for primes in arithmetical progressions due to mccurley 6 and obtain the following quantitative result theorem let a b be integers with 0 b a define d by 0 if 3 4 a b 3 d i1 if 311 a b 13 2 otherwise further define a k c and q0 by 4 x 2328 3d al1 2 b1 2 2 received may 9 1985 1980 mathematics subject classification primary iof25 1986 american mathematical society 0025 5718 86 1 00 25 per page 613 this content downloaded from 207 46 13 124 on wed 22 jun 2016 05 36 01 utc all use subject to http about jstor org terms