distribution of values of bounded generalized polynomials a generalized polynomial is a real valued function which is obtained from conventional polynomials by the use of the operations of addition multiplication and taking the integer part a generalized polynomial mapping is a vector valued mapping whose coordinates are generalized polynomials we show that any bounded generalized polynomial mapping u zd rl has a representation u n f n x n zd where f is a piecewise polynomial function on a compact nilmanifold x x x and is an ergodic zd action by translations on x this fact is used to show that the sequence u n n zd is well distributed on a piecewise polynomial surface mathcal s subset mathbf r l with respect to the borel measure on mathcal s that is the image of the lebesgue measure under the piecewise polynomial function defining mathcal s as corollaries we also obtain a von neumann type ergodic theorem along generalized polynomials and a result on diophantine approximations extending the work of van der corput and of furstenberg weiss

on bootstrap iteration for coverage correction in confidence intervals abstract iterated bootstrap procedures may be used to reduce error in many statistical problems we discuss their use in constructing confidence intervals with accurate coverage and show that bootstrap coverage correction produces improvements in coverage accuracy of order n 1 2 in one sided intervals but of order n 1 in two sided intervals explicit formulas are provided for the dominant term in coverage error after iteration in each case these results are used to compare various iterated bootstrap intervals and to assess the effect of bootstrap iteration on other indicators of interval performance such as position of critical points and length of interval we show that for one sided intervals the coverage correction algorithm yields critical points that are second order correct this is not the case for two sided intervals where second order correctness is not crucial in obtaining high order coverage accuracy we also show that the asymptotic mean increase in length between the original and cove

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

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

nonconventional ergodic averages and nilmanifolds we study the l2 convergence of two types of ergodic averages the first is the average of a product of functions evaluated at return times along arithmetic progressions such as the expressions appearing in furstenberg fs proof of szemer ledi fs theorem the second average is taken along cubes whose sizes tend to for each average we show that it is sufficient to prove the convergence for special systems the characteristic factors we build these factors in a general way independent of the type of the average to each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold from the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density

the spectrum of a coxeter transformation affine coxeter transformations and the defect map abstract the spectrum of the coxeter transformation r in a reflection group defined by a matrix a without odd cycles is determined in terms of spec a for affine a r determines the defect map which is given a simple formulation in terms of null roots the affine exponents determined by spec r are related to the coefficients of the fundamental co weight corresponding to the fork node of the coxeter dynkin diagram

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

bootstrap bartlett adjustment in seemingly unrelated regression abstract seemingly unrelated regression sur is a method introduced by zellner 1962 for estimating several regression equations simultaneously a common and important problem in econometrics this article considers hypothesis testing in sur using the log likelihood ratio llr test although the asymptotic distribution of this statistic is well known substantial departures occur in samples of the size commonly employed by economists significance levels given by the use of this asymptotic distribution are too low sometimes by an order of magnitude leading to the rejection of too many true null hypotheses a common approach to this problem is to use a bartlett 1937 adjustment to the llr statistic in which the test statistic is multiplied by a factor derived from second order asymptotics before it is referred to the asymptotic 2 q distribution this article proposes that the bartlett adjustment factor be computed using efron s bootstrap 1979 1982 in the cases examined this leads to more accu

the saddlepoint correction in conditional logistic likelihood analysis summary barndorff nielsen cox 1979 introduced the double saddlepoint approximation to the conditional likelihood function for a structural parameter vector given a sufficient statistic for a nuisance parameter vector this approximation is considered in some detail for the case of multiple logistic regression models with polytomous outcome states it is shown that the profile score with saddlepoint correction yields a simple derivation of an approximation to the mean of a noncentral hypergeometric random variable discussed by levin 1984 and gart 1987 while providing analogous formulae in more general cases and extends to the general case an algorithm of mccullagh 1984 for the approximate calculation of conditional maximum likelihood estimates of log odds ratio regression coefficients from several fourfold tables

intersective polynomials and the polynomial szemeredi theorem abstract let p p 1 p r q n 1 n m be a family of polynomials such that p i z m z i 1 r we say that the family p has the psz property if for any set e z with d e lim sup n m e m n 1 n m 0 there exist infinitely many n z m such that e contains a polynomial progression of the form a a p 1 n a p r n we prove that a polynomial family p p 1 p r has the psz property if and only if the polynomials p 1 p r are jointly intersective meaning that for any k n there exists n z m such that the integers p 1 n p r n are all divisible by k to obtain this result we give a new ergodic proof of the polynomial szemeredi theorem based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds we also obtain as a corollary the following generalization of the polynomial van der waerden theorem if p 1 p r q n are jointly intersective integral polynomials then for any finite partition z i 1 k e i of z there exist i 1 k and a n e i such that a a p 1 n a p r n e i

ipr recurrence and nilsystems abstract by a result due to furstenberg a homeomorphism t of a compact space is distal if and only if it possesses the property of ip recurrence meaning that for any x 0 x for any open neighborhood u of x 0 and for any sequence n i in z the set r u x 0 n z t n x 0 u has a non trivial intersection with the set of finite sums n i 1 n i 2 n i s i 1 i 2 i s s n we show that translations on compact nilmanifolds which are known to be distal are characterized by a stronger property of ip r recurrence which asserts that for any x 0 x and any neighborhood u of x 0 there exists r n such that for any r element sequence n 1 n r in z the set r u x 0 has a non trivial intersection with the set n i 1 n i 2 n i s i 1 i 2 i s s r we also show that the property of ip r recurrence is equivalent to an ostensibly much stronger property of polynomial ip r recurrence this should be juxtaposed with the fact that for general distal transformations the polynomial ip recurrence is strictly stronger than the ip recurrence

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

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

saddlepoint approximations for likelihood ratio like statistics with applications to permutation tests we obtain two theorems extending the use of a saddlepoint approximation to multiparameter problems for likelihood ratio like statistics which allow their use in permutation and rank tests and could be used in bootstrap approximations in the first we show that in some cases when no density exists the integral of the formal saddlepoint density over the set corresponding to large values of the likelihood ratio like statistic approximates the true probability with relative error of order l n in the second we give multivariate generalizations of the lugannani rice and barndorff nielsen or r formulas for the approximations these theorems are applied to obtain permutation tests based on the likelihood ratio like statistics for the k sample and the multivariate two sample cases numerical examples are given to illustrate the high degree of accuracy and these statistics are compared to the classical statistics in both cases

stochastic network models for survival analysis abstract we present methodology giving highly accurate approximations for bayesian predictive densities and distribution functions of first passage times between states of a semi markov process with a finite number of states when the states describe a degenerative disorder with an absorbing end state such predictive distributions are the survival distributions of a patient we illustrate these methods with a variety of examples including data from the san francisco aids study we achieve our approximations using a three step sequence first we introduce advanced concepts of flowgraph theory which allow us to compute the moment generating function of the first passage time given the model parameters next we use saddlepoint approximations to convert this into a density or distribution function conditional on the model parameter finally we use monte carlo methods to remove dependence on the model parameter these methods apply quite generally to all finite state semi markov models in discrete or con

killing and the coxeter transformation of kac moody algebras on etudie le spectre de la transformation de coxeter associee a une algebre de kac moody arbitraire et plus particulierement a une algebre affine

on the relationship between bayesian and non bayesian interval estimates summary let y1 y denote independent observations each distributed according to a density depending on a scalar parameter 0 suppose that we are interested in constructing an interval estimate for 0 one approach is to construct a confidence region with a specified coverage probability based on the likelihood ratio test statistic another approach is to construct a highest posterior density region with a specified posterior probability the goal of this paper is to study the relationship between these two approaches in particular we derive asymptotic expansions for the posterior probability of confidence regions based on the likelihood ratio test statistic and for the coverage probability of highest posterior density regions conditions under which the two methods lead to identical regions at least approximately are also given

smooth and accurate multivariate confidence regions this article describes multivariate approximate conditional confidence regions for canonical exponential families these confidence regions have actual coverage probabilities that are closer to their nominal levels than are the actual coverage probabilities of traditional normaltheory regions and have boundaries that are smoother than those obtained by inverting traditional exact tests our method is based on constructing one dimensional conditional tests combining p values and inverting more specifically consider a statistical model with three parameters of which two are of interest and one is not of interest we generate a confidence region for the two parameters of interest by first generating a confidence interval for one of the parameters conditional on sufficient statistics associated with the other interest parameter and the nuisance parameter for values of this bounded parameter inside the confidence interval we determine a confidence interval for the remaining interest parameter condition

improved likelihood ratio statistics for exponential family nonlinear models summary this paper gives a general formula for the expected likelihood ratio statistic for exponential family nonlinear models corrected up to order n 1 where n is the sample size this result generalizes previous work by cordeiro 1983 various general bartlett adjustment formulae are given to some special models and improved likelihood ratio tests are discussed

a bartlett adjustment to the likelihood ratio test for a system of equations abstract a bartlett adjustment to be applied to the likelihood ratio test statistic when testing parameter restrictions in a system of equations is derived as special cases results are also obtained for testing hypotheses on the mean of a normal variate linear restrictions in the linear model restrictions in a nonlinear model and linear restrictions in the multivariate linear model

certain imprimitive reflection groups and their generic versions the present paper is concerned with the connection between the imprimitive reflection groups g m m n m n and the affine weyl group a n 1 we show that a n 1 is a generic version of the groups g m m n m n we introduce some new presentations of these groups which are shown to have some group theoretic advantages then we define the hecke algebras of these groups and of their braid versions each in two ways according to two presentations finally we give a new description for the affine root system of a n 1 such that the action of a n 1 on is compatible with that of g m m n on its root system in some sense

on conditioning for model based inference in survey sampling summary a discontinuity of sampling distribution inference between a prediction problem and an estimation problem in model based survey sampling is discussed the problem is resolved by conditioning on an appropriate ancillary within a curved exponential family framework the ignorability of the sampling design for inference is discussed

on the level error after bartlett adjustment of the likelihood ratio statistic summary it is well known that bartlett adjustment reduces level error of the likelihood ratio statistic from order n l to order n3 2 in the present note we show that level error of the adjusted statistic is actually order n2

likelihood inference for linear regression models summary approximate conditional inference based on large sample likelihood ratio methods is considered for the parameters of linear regression models mean and variance adjustments that improve the standard normal approximation to the conditional distribution of the signed square root of the log likelihood ratio statistic for a scalar parameter of interest are given a bartlett adjustment factor that improves the chi squared approximation to the conditional distribution of the log likelihood ratio statistic for a vector parameter of interest is also presented the accuracy of approximate confidence limits obtained by using the adjustments is demonstrated for a location scale analysis of darwin s data

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

double exponential families and their use in generalized linear regression abstract in one parameter exponential families such as the binomial and poisson the variance is a function of the mean double exponential families allow the introduction of a second parameter that controls variance independently of the mean double families are used as constituent distributions in generalized linear regressions in which both means and variances are allowed to depend on observed covariates the theory is applied to two examples a logistic regression and a large two way contingency table in such cases the binomial model of variance is often untrustworthy for example because genuine random sampling was infeasible the subjects may have been obtained in clumps so that the statistician should really be using smaller sample sizes clumped sampling is just one of many possible causes of overdispersion a habitual source of concern to users of binomial and poisson models this article concerns a class of regression families that allow the statistician to model overdispersion while carrying

set polynomials and polynomial extension of the hales jewett theorem an abstract hales jewett type extension of the polynomial van der waerden theorem bl is established theorem let r d q n there exists n n such that for any r coloring of the set of subsets of v 1 n d 1 q there exist a set a v and a nonempty set 1 n such that a d 1 q and the subsets a a d 1 a d 2 a d q are all of the same color this polynomial hales jewett theorem contains refinements of many combinatorial facts as special cases the proof is achieved by introducing and developing the apparatus of set polynomials polynomials whose coefficients are finite sets and applying the methods of topological dynamics

improved likelihood ratio statistics for generalized linear models summary this paper gives a general formula for the expected value of the likelihood ratio statistic for generalized linear models corrected up to terms of order n 1 where n is the size of the sample the application of this formula to some models is illustrated

is the improved likelihood ratio statistic really improved in the discrete case summary on the basis of some numerical studies we argue that bartlett adjustment of the likelihood ratio statistic for discrete data does not in general yield an improvement in the asymptotic error rate of the chi squared approximation also we argue that the small sample effect of the bartlett adjustment is unclear and that it is in no way comparable with the improvement obtained for continuous data

a note on bartlett adjustments to likelihood ratio tests summary the invariance properties of the likelihood ratio statistic are used to question the validity of certain terms arising in the expansion of its null distribution it is suggested that the null distribution of bartlett adjusted tests is the appropriate reference chi squared distribution with error of order n 3 2 where n is the sample size bartlett adjustment factors are a widely used method for improving the large sample chi squared approximation to the null distribution of maximum likelihood ratio statistics among the major developments into the calculation and effect of the correction factors are those due to lawley 1956 hayakawa 1977 and barndorff nielsen cox 1984 for regular problems lawley showed that all of the cumulants of the bartlett adjusted log likelihood ratio statistic for testing a composite hypothesis agree with those of the appropriate chi squared reference distribution with error of order n 3 2 where n is the sample size assumed large hayakawa obtained an asymptotic expansion to order n l for the null distribution of the test statistic considered by lawley and obtained a slightly different result following barndorff nielsen cox 1984 let hayakawa s expansion for the null distribution of the likelihood ratio statistic for testing a composite hypothesis be written as f x qd x 24n f a2qd 4 x 2a2 al qd 2 x a2 al qd x o n 3 2 1 here qd x denotes the density function of the chi squared distribution with d degrees of freedom d being the dimension of the vector parameter specified under the null hypothesis and the terms al and a2 given in 7 of hayakawa are functions of the moments of certain derivatives of the log likelihood function for the special case d 1 and a single nuisance parameter the term a2 expressed in a particularly convenient form given in the imperial college thesis of g m cordeiro is also given here in 2 note that p 1 12dn fal is the bartlett adjustment factor lawley s result corresponds to that in 1 with the term a2 taken to be zero and in many commonly occurring situations the term a2 as given by hayakawa when calculated explicitly is in fact zero our purpose here is to suggest that there is an error in the expansion due to hayakawa and that in 1 the term a2 should not appear making hayakawa s result explicitly agree with that of lawley the basis of the argument is presented in the next section

improved likelihood ratio statistics for covariance selection models summary the expected value of the likelihood ratio statistic for covariance selection models when this statistic has a closed form is derived correct to terms of order n 2 n being the sample size then a bartlett type correction factor is available for such statistics by simulation the practical use of such a correction factor is illustrated and is compared with an alternative correction factor in a special case the null density of the corrected statistic is also considered

bartlett s cochran s and hartley s tests on variances are liberal when the underlying distribution is long tailed abstract for samples of equal size the p values of bartlett s cochran s and hartley s tests for the equality of several variances obtained under the normality assumption are shown to underestimate the true p values when the parent distribution is a normal or a x 2 1 scale mixture or when there is within sample dependence the proof uses elements of majorization theory when the sample sizes are not equal a partial extension is obtained for the two sample case

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

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

the mobius function is strongly orthogonal to nilsequences we show that the mobius function n is strongly asymptotically or thogonal to any polynomial nilsequence f g n n2n here g is a sim ply connected nilpotent lie group with a discrete and cocompact subgroup so g is a nilmanifold g z g is a polynomial sequence and f g r is a lipschitz function more precisely we show that j 1 n p n n 1 n f g n j f g a log a n for all a 0 in particu lar this implies the mobius and nilsequence conjecture mn s from our earlier paper for every positive integer s this is one of two major in gredients in our programme to establish a large number of cases of the generalised hardy littlewood conjecture which predicts how often a collec tion 1 t z d z of linear forms all take prime values the proof is a relatively quick application of the results in our recent companion paper we give some applications of our main theorem we show for exam ple that the mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c we also obtain a result about the distribution of nilse quences a n x n2n as n ranges only over the primes

maximum likelihood estimation and large sample inference for generalized linear and nonlinear regression models summary the class of generalized linear models is extended to allow for correlated observations nonlinear models and error distributions not of the exponential family form the extended class of models include a number of important examples particularly of the composite transformational type large sample inference and maximum likelihood estimation for the extended class of generalized linear models are discussed and the analysis of deviance is generalized to the extended class of models calculation of the maximum likelihood estimate for a general likelihood by fisher s scoring method and a related method is considered and the relation with the gauss newton method is discussed

saddlepoint approximations and tests based on multivariate m estimates we consider multidimensional m functional parameters defined by expectations of score functions associated with multivariate m estimators and tests for hypotheses concerning multidimensional smooth functions of these parameters we propose a test statistic suggested by the exponent in the saddlepoint approximation to the density of the function of the m estimates this statistic is analogous to the log likelihood ratio in the parametric case we show that this statistic is approximately distributed as a chi squared variate and obtain a lugannani rice style adjustment giving a relative error of order we propose an empirical exponential likelihood statistic and consider a test based on this statistic finally we present numerical results for three examples including one in robust regression

robust methods for testing the pattern of a single covariance matrix summary patterns for a single covariance matrix may be studied using likelihood ratio or quadratic form test statistics typically these tests have an asymptotic chi squared distribution when the data are normally distributed but the asymptotic distribution is sensitive to nonnormality this paper suggests two ways to robustify inference for patterned matrices i estimate the critical values of normal theory test statistics using a new bootstrap procedure which takes advantage of certain invariances under ho and ii modify the statistics so that an asymptotic chi squared distribution holds even under nonnormality

a decomposition for the likelihood ratio statistic and the bartlett correction a bayesian argument let l n 1 log p x be the log likelihood of an n dimensional x under a p dimensional let j be the mle under h j 1 1 0 j j 0 and 0 be the unrestricted mle define t j as 2n l j 1 1 2 sgn j j 1 j 0 let t t 1 t p then under regularity conditions the following theorem is proved under 0 t is asymptotically n n 1 2 a 0 n 1 a j n 1 o n 3 2 where j is the identity matrix the result is proved by first establishing an analogous result when is random and then making the prior converge to a degenerate distribution the existence of the bartlett correction to order n 3 2 follows from the theorem we show that an edgeworth expansion with error o n 2 for t involves only polynomials of degree less than or equal to 3 and hence verify rigorously lawley s 1956 result giving the order of the error in the bartlett correction as o n 2

nilsequences and a structure theorem for topological dynamical systems abstract we characterize inverse limits of nilsystems in topological dynamics via a structure theorem for topological dynamical systems that is an analog of the structure theorem for measure preserving systems we provide two applications of the structure the first is to nilsequences which have played an important role in recent developments in ergodic theory and additive combinatorics we give a characterization that detects if a given sequence is a nilsequence by only testing properties locally meaning on finite intervals the second application is the construction of the maximal nilfactor of any order in a distal minimal topological dynamical system we show that this factor can be defined via a certain generalization of the regionally proximal relation that is used to produce the maximal equicontinuous factor and corresponds to the case of order 1

diagnostics for assessing the accuracy of normal approximations in exponential family nonlinear models abstract diagnostics are investigated for assessing the agreement between likelihood and standard large sample confidence regions for parameters from an exponential family nonlinear model cordeiro and paula 1989 the development connects the contour methods proposed by hodges 1985 1987 with the curvature measures for normal nonlinear regression proposed by bates and watts 1980 and jennings s 1982 1986 diagnostics for logistic regression the proposed methodology is illustrated with several examples

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

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

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

conjugacy relation on coxeter elements abstract let w s be an irreducible finitely presented coxeter system the present paper is mainly concerned with a conjugacy relation on coxeter elements in the case where contains just one circle in particular when is itself a circle in the cases where is either a three multiple circle or a circle with three nodes we show that the ss equivalence relation on coxeter elements of w is the same as the w conjugacy relation an explicit formula is given for the characteristic polynomial of a coxeter element in the natural reflection representation of w when is a circle we also give the answers to some questions raised by coleman and extend some results of geck and pfeiffer concerning conjugacy relation in coxeter groups

a second order investigation of asymptotic ancillarity the paper deals with approximate ancillarity as discussed by efron and hinkley 1978 in the multivariate i i d case we derive the second order edgeworth expansion of the mle given a normalized version of the second derivative of the log likelihood at its maximum the expansion agrees with the one derived by amari 1982a for curved exponential families but holds for any family satisfying the regularity conditions given in the paper it is shown that the fisher information lost by reducing the data to the mle is recovered by the conditioning and it is sketched how the loss of information relates to the deficiency as defined by lecam finally we investigate some properties of three test statistics proving a conjecture by efron and hinkley 1978 concerning the conditional null distribution of the likelihood ratio test statistic and establishing a kind of superiority of the observed fisher information over the expected one as estimate of the inverse variance of the mle

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

effective degrees of freedom and the likelihood ratio test summary normal theory problems involving several unknown variances may require use of estimates not having simple chi squared distributions confidence intervals are then commonly based on the use of effective degrees of freedom i e the fitting of a gamma distribution via the first two moments replacing parameters by estimates the confidence intervals given by this method are compared with those resulting from the likelihood ratio test inserting a bartlett adjustment factor to achieve close approximation by a chi squared distribution two special problems one concerning components of variance and one a generalization of the behrens fisher problem are studied in detail

a comparison of asymptotically equivalent test statistics for regression transformation summary a simulation comparison is made of six asymptotically equivalent test statistics for transformation of the response in regression the tests typically have observed sizes up to twice the nominal size a likelihood ratio statistic a score statistic and a wald statistic behave very similarly both at the null hypothesis and in terms of power values of all the statistics are highly positively correlated the greater part of any differences between the statistics is concentrated in their standard deviations to test hypotheses about parametric families of transformation box cox 1964 use a likelihood ratio test more recently several other asymptotically equivalent tests have been suggested in the present paper we present empirical comparisons of six tests in terms of size power and diagnostic efficacy the interesting relevance of these results to small sample comparisons of score wald and likelihood ratio tests is briefly discussed the notation and background of the paper follow those of box cox 1964 a vector of n response values y is linearly related with an initial constant term to p 1 explanatory variables whose values and an initial column of ones are contained in an n xp matrix x the box cox family of transformations indexed by a scalar parameter a is employed to produce the transformed response variable y the aim is to choose a by maximum likelihood so that y follows the linear model y a x 3 e 1 1 where e is a vector of errors which are mutually independent with normal distributions of zero mean and constant variance a2 and 3 is a vector of coefficients the estimate a satisfies the equation

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

invariants and likelihood ratio statistics because the likelihood ratio statistic is invariant under reparameterization it is possible to make a large sample expansion of the statistic itself and of its expectation in terms of invariants in particular the bartlett adjustment factor can be expressed in terms of invariant combinations of cumulants of the first two log likelihood derivatives such expansions are given first for a scalar parameter and then for vector parameters geometrical interpretation is given where possible and some special cases discussed

a note on the calculation of bartlett adjustments summary we describe for the numerical calculation of bartlett adjustments a method which may be of use when the cumulants of the log likelihood derivatives are easy to determine in one parametrization while the hypotheses to be tested are all linear in some other parametrization the method relies on the cartesian tensorial nature of those cumulants and should be particularly convenient in connection with statistical packages with structure similar to glim

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

fully commutative elements in the weyl and affine weyl groups abstract let w be a weyl or an affine weyl group and let w c be the set of fully commutative elements in w we associate each w w c to a digraph g w by using g w we give a graph theoretic description for lusztig s a function on w c and describe explicitly all the distinguished involutions of w c the results verify two conjectures in our case one was proposed by myself in adv sci china math 3 1990 79 98 conjecture 8 10 and the other by lusztig in t asai et al open problems in algebraic groups in r hotta ed problems from the conference on algebraic groups and representations katata august 29 september 3 1983

on parameter transformations and interval estimation summary parameterizations which reduce the asymptotic bias and skewness of various pivotal quantities arising in large sample theory are discussed for models depending on an unknown scalar parameter transformation formulae by which such parameterizations can be obtained are derived and these formulae extend those for one dimensional curved exponential families given by hougaard 1982 to assess the accuracy of normal approximations to the distributions of the pivots their second order properties are considered and comparisons with the signed square root of the likelihood ratio statistic are drawn

the removal method for estimating natural populations incorporating auxiliary information the removal method was originally designed to estimate the size of a pest population the technique involves monitoring the decline in the success rate in catching animals as the population is reduced if for example after removing 25 rats from a population the success rate in catching further rats were to drop by a factor of 2 one might guess that the population had declined to half its original size this is the reasoning behind the basic removal estimate in this instance the population size would be estimated at 50 individuals of course it is also possible that the rat population is substantially larger and that the large decline in numbers captured is attributable to another factor the remaining rats for example could be better at avoiding capture thus the removal method like most other techniques for estimating parameters of natural populations is plagued by assumptions that are at best highly questionable furthermore these methods often ignore potentially valuable auxiliary evidence signs of activity such as scats etc can also provide clues as to the size of the remaining population methodology intended to combat simultaneously both these deficiencies is proposed here the basic notion is that when the input for an estimation scheme is extended to include information of more than one type the dependence on the restrictive assumptions associated with the use of any single component will be reduced the statistical methodology involves a standard computer calculation of a maximum likelihood estimate and a slight modification to a common scheme for generating confidence limits the chi squared approximation to the distribution of the maximum likelihood ratio test statistic monte carlo simulations are used to demonstrate that with the inclusion of information on signs of activity the estimation scheme becomes more robust to departures from the assumptions of the removal method

exponential empirical likelihood is not bartlett correctable in a recent paper diciccio hall and romano established that owen s empirical likelihood is bartlett correctable this is an intriguing and perhaps surprising result and is the only nonparametric context in which bartlett correctability is known to hold an alternative closely related nonparametric likelihood referred to here as exponential empirical likelihood may be constructed using efron s method of nonparametric tilting the purpose of this note is to show that exponential empirical likelihood is not bartlett correctable

prepivoting test statistics a bootstrap view of asymptotic refinements abstract approximate tests for a composite null hypothesis about a parameter may be obtained by referring a test statistic to an estimated critical value either asymptotic theory or bootstrap methods can be used to estimate the desired quantile the simple asymptotic test a refers the test statistic to a quantile of its asymptotic null distribution after unknown parameters have been estimated the bootstrap approach used here is based on the concept of prepivoting prepivoting is the transformation of a test statistic by the cdf of its bootstrap null distribution the simple bootstrap test b refers the prepivoted test statistic to a quantile of the uniform 0 1 distribution under regularity conditions the bootstrap test b has a smaller asymptotic order of error in level than does the asymptotic test a provided that the asymptotic null distribution of the test statistic does not depend on unknown parameters in the contrary case both a and b have the same order of level error certain class

a modified score test statistic having chi squared distribution to order n 1 summary we give a general modified score test statistic whose nuil distribution is chi squared to order n 1 where n is the sample size the modified statistic depends on the joint cumulants of log likelihood derivatives for the full data some applications are discussed following cox reid 1987a we also derive a general formula for bartlett type corrections to improve test statistics whose null asymptotic distributions are chi squared

pseudo likelihood theory for empirical likelihood it is proved that except for a location term empirical likelihood does draw contours which are second order correct for those of a pseudo likelihood however except in the case of one dimension this pseudo likelihood is not that which would commonly be employed when constructing a likelihood based confidence region it is shown that empirical likelihood regions may be adjusted for location so as to render them second order correct furthermore it is proved that location adjusted empirical likelihood regions are bartlett correctable in a certain sense

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

an empirical adjustment to the likelihood ratio statistic consider a model parameterised by a scalar parameter of interest and a nuisance parameter inference about may be based on the signed square root of the likelihood ratio statistic r the statistic r is asymptotically distributed according to a standard normal distribution with error o n 1 2 to reduce the error of this normal approximation several modifications to r have been proposed such as barndorff nielsen s modified directed likelihood statistic r in this paper an approximation to r is proposed that can be calculated numerically for a wide range of models this approximation is shown to agree with r with error of order o p n 1 the results are illustrated on several examples