simplicity and superrigidity of twin building lattices kac moody groups over finite fields are finitely generated groups most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non positively curved twinned buildings those are the most important but not the only examples of twin building lattices we prove that these lattices are simple if the corresponding buildings are irreducible and not of affine type i e they are not bruhat tits buildings many of them are finitely presented and enjoy property t our arguments explain geometrically why simplicity fails to hold only for affine kac moody groups moreover we prove that a nontrivial continuous homomorphism from a completed kac moody group is always proper we also show that kac moody lattices fulfill conditions implying strong superrigidity properties for isometric actions on non positively curved metric spaces most results apply to the general class of twin building lattices

quadratic zeros of racah 6j coefficients a geometrical approach it is shown that the projective symmetries of the polynomial of quadratic 6j coefficients form the symmetrical group s6 nonlinear rational symmetries of are found partial parametrizations of the zeros of are presented

raising and lowering operators factorization and differential difference operators of hypergeometric type starting from the rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variables on a uniform lattice in order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight using the infeld hull factorization method we generate from the raising and lowering operators the second order self adjoint differential difference operator of hypergeometric type

a complete determination of the zeros of weight 1 6j coefficients it is shown how all zeros of weight 1 6j coefficients arise as particular cases of a four parameter family of such zeros the parametrization is given explicitly

representations of the discrete inhomogeneous lorentz group and dirac wave equation on the lattice we propose the fundamental and two dimensional representation of the lorentz groups on a 3 1 dimensional hypercubic lattice from which representations of higher dimensions can be constructed for the unitary representation of the discrete translation group we use the kernel of the fourier transform from the dirac representation of the lorentz group including reflections we derive in a natural way the wave equation on the lattice for spin 1 2 particles finally the induced representation of the discrete inhomogeneous lorentz group is constructed by standard methods and its connection with the continuous case is discussed

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

stable and invariant adjusted profile likelihood and directed likelihood for curved exponential models summary for curved exponential models the modified profile likelihood and the modified directed likelihood are approximated with error o n 1 under ordinary repeated sampling by explicitly defined and invariant quantities which do not require specification of an ancillary and which are stable in the sense of having the appropriate conditionality property with respect to any reasonable ancillary

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

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

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

raising and lowering operators and their factorization for generalized orthogonal polynomials of hypergeometric type on homogeneous and non homogeneous lattices we complete the construction of raising and lowering operators given in a previous work for the orthogonal polynomials of hypergeometric type on a non homogeneous lattice we extend these operators to the generalized orthogonal polynomials namely those difference orthogonal polynomials that satisfy a similar difference equation of hypergeometric type

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

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

adjusted versions of profile likelihood and directed likelihood and extended likelihood several adjusted and invariant versions of profile likelihood and of directed likelihood short for signed log likelihood ratio are discussed these adjusted versions which fall into two groups are closely related to modified profile likelihood and modified directed likelihood the first group has the advantage of not requiring an explicit specification of an ancillary but partly at a cost of a loss of asymptotic accuracy in comparison with the modified versions the second group consists of some asymptotically sharpened versions of modified profile likelihood furthermore a concept of extended likelihood is proposed which associates a likelihood like function to certain combinants

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

on the zeros of 6 j coefficients an extensive computer search for non trivial zeros of the 6 j coefficients of su 2 has been performed to find their frequency as a function of the sum of their quantum numbers and of their polynomial degree for the zeros of degree 2 four of the seven cases already known taking into account regge symmetries of two relations between quantum numbers relevant to a pell equation have been generalized to deal with only one relation a new case of two relations and many new cases of three relations all with polynomial solutions have been found but all these formulae only account for one zero out of seven for degree 3 we found four new cases of two relations and some other cases of three relations with polynomial solutions with the already known case of a pell equation they account for almost half of the zeros the fast decrease of the number of zeros with the degree and the relative smallness of their quantum number is an indication that there should be no zeros of degree larger than 9

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

curtis tits groups generalizing kac moody groups of type a n 1 abstract in 13 we define a curtis tits group as a certain generalization of a kac moody group we distinguish between orientable and non orientable curtis tits groups and identify all orientable curtis tits groups as kac moody groups associated to twin buildings in the present paper we construct all orientable as well as non orientable curtis tits groups with diagram a n 1 n 4 over a field k of size at least 4 the resulting groups are quite interesting in their own right the orientable ones are related to drinfeld s construction of vector bundles over a non commutative projective line and to the classical groups over cyclic algebras the non orientable ones are related to expander graphs 14 and have symplectic orthogonal and unitary groups as quotients

likelihood for component parameters for a statistical model with data likelihood for the scalar or vector full parameter 0 of dimension p say is typically well defined and easily computed in this paper we investigate likelihood for a component parameter 0 of dimension d p and make use of the recent likelihood theory that has been successful in producing highly accurate third order p values for scalar parameters of continuous models the theory leads under moderate regularity to a definitive third order determination of likelihood for a component parameter 0 of dimension d where 1 d p we use the simple location model on the plane with standard normal errors to motivate the development the example exhibits most of the key characteristics of the general case and the recent theory then extends the determination of likelihood to the general context for the scalar interest parameter case with d 1 the usual determinations are typically of second order accuracy the example indicates how the new determination achieves third order accuracy the implementation is straightforward and uses familiar ingredients to other determinations such as the full maximum likelihood value o the constrained value o given 0 and the observed information j o for a complementing nuisance parameter 0 it does however require a special version of the nuisance information j o a version calibrated relative to a symmetric choice of the exponential type reparameterisation 0 underlying the recent theory but this is easily computed various examples are given and the motivating example is discussed in detail

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

nontrivial zeros of the wigner 3j and racah 6j coefficients ii some nonlinear solutions additional formulas for nontrivial zeros in the 3j and 6j symbols have been found for some higher order cases i e where k 2 3 and 4 numerical examples only

on lie algebras built from so 3 tensor operators concerning the lie algebras generated by certain linear combinations of so 3 tensor operators two theorems are given which disprove some widely accepted results in the standard literature

1 twinnings of buildings in this paper we characterize twinnings of buildings by 1 twinnings and one further condition concerning twin apartments specialized to the spherical case we obtain new characterizations of the opposition relation in such buildings we also give a new description of the standard twin building of type widetilde a n 1

adjoint boson realization of su n and a family of structural zeros of 6j coefficients for the fundamental boson realization of su n in its principal so 3 basis the generators of su n are so 3 tensor operators constructed out of one single type of boson namely a boson with so 3 multiplet label n 1 2 here the adjoint boson realization of su n is given this time the so 3 tensor operators are constructed by means of n 1 types of bosons those with so 3 labels 1 2 n 1 the validity of this realization depends upon a new identity between 6j coefficients from this realization or equivalently from this new identity one can deduce the existence of a family of structural zeros of 6j coefficients

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

continuum quantum systems as limits of discrete quantum systems iv affine canonical transforms affine canonical transforms complex order fourier transforms and their associated coherent states appear in two scenarios finite discrete and continuum we examine the relationship between the two scenarios making systematic use of inductive limits which were developed in the preceding articles in this series

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

weight 2 zeros of 3j coefficients and the pell equation all weight 2 zeros of the wigner 3j coefficients may be obtained from the quadratic diophantine equation known as pell s equation these zeros may then be classified by the orbits of a discrete infinite order subgroup of the lorentz group so 1 1 this is carried out by transforming the polynomial part of a weight 2 3j coefficient to pellian form and obtaining the fundamental zeros numerically the relation of this polynomial to a family of binary quadratic forms is also given together with a discussion of the invariance group

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

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

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

codistances of 3 spherical buildings we show that a 3 spherical building in which each rank 2 residue is connected far away from a chamber and each rank 3 residue is simply 2 connected far away from a chamber admits a twinning i e is one half of a twin building as soon as it admits a codistance i e a twinning with a single chamber

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

nontrivial zeros of the wigner 3 j and racah 6 j coefficients i linear solutions some formuals for nontrivial zeros in the 3 j and 6 j symbols have been found some formuals for nontrivial zeros in the 3 j and 6 j symbols have been found

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

likelihood functions for inference in the presence of a nuisance parameter consider inference about a scalar parameter of interest in the presence of a vector nuisance parameter inference about is often based on a pseudolikelihood function in this paper the general problem of constructing a pseudo loglikelihood function h is considered conditions are given under which h has the same properties as a genuine loglikelihood function for a model without a nuisance parameter when these conditions are satisfied to a given order of approximation h is said to be a jth order local loglikelihood function the theory of local loglikelihood functions is developed and it is shown that second order versions of these have a number of desirable properties several commonly used pseudolikelihood functions are studied from this point of view one commonly used pseudolikelihood function is profile likelihood in which parameters other than 0 are replaced by their maximum likelihood estimates a second aspect of the paper is to consider the use of other estimates in this context examples are given which suggest that inference about may be improved if a method other than maximum likelihood is used particularly when the number of other parameters is large relative to the sample size

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

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

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

parametrization of the linear zeros of 6j coefficients the linear zeros of 6j coefficients are fully parametrized apart from a multiplicative factor in terms of four integers

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

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

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

on diophantine equations and nontrivial racah coefficients some families of zeros of weight 1 6j coefficients are given each in terms of four parameters they arise from a geometrical investigation of certain diophantine equations some general remarks on the solutions of diophantine equations are also made

nontrivial zeros of the racah quadrupole invariant it is shown that a class of nontrivial zeros of the racah quadrupole invariant operator is given by two orbits of the group action of an infinite discrete subgroup of the proper two dimensional lorentz group so 1 1 on the hyperbola 4x2 3y2 11 4

a new parametrization and all integral realizations of the lorentz group a simple linear transformation of the biquaternionic parameters and of the group sl 2 c which is two to one homomorphic to the restricted lorentz group l is used to express each element of sl 2 c and l in terms of the first column of an element of l and the quaternionic parameters this parametrization is shown to bring the determination of the integral realizations of the restricted lorentz group under the purview of classical diophantine analysis involving the expression of a given pair of integers respectively as a sum of three squares and a sum of four squares these solutions are further constrained by the condition that a set of four integers which is linear in the solution of the three squares problem and in the solution of the four squares problem have a common factor these are all classical problems addressed and solved by euclid euler fermat gauss jacobi and others the corresponding realizations of sl 2 c fall into three distinct classes those in which the elements of

tensor product of group representations and structural zeros of racah coefficients it is shown that structural or nontrivial zeros of racah coefficients 6 j symbols can be related to decompositions of tensor products of group representations reduced to so 3 this extends a previous relation between structural zeros of 6 j symbols and boson realizations of exceptional lie groups in the present case both classical and exceptional groups may appear and examples are given for su 3 so 7 so 9 and f4

nontrivial zeros of weight 1 3j and 6j coefficients relation to diophantine equations of equal sums of like powers the nontrivial zeros of weight 1 3j and 6j coefficients given previously are shown to be the set of all such zeros the relation of these zeros to the solutions of well known diophantine equations is also discussed

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

the double torus as a 2d cosmos groups geometry and closed geodesics the double torus provides a relativistic model for a closed 2d cosmos with topology of genus 2 and constant negative curvature its unfolding into an octagon extends to an octagonal tessellation of its universal covering the hyperbolic space h2 the tessellation is analysed with tools from hyperbolic crystallography actions on h2 of groups subgroups are identified for su 1 1 for a hyperbolic coxeter group acting also on su 1 1 and for the homotopy group 2 whose extension is normal in the coxeter group closed geodesics arise from links on h2 between octagon centres the direction and length of the shortest closed geodesics is computed dedicated to marcos moshinsky on the occasion of his 80th birthday

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

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