geometric formulation of mechanical systems with one sided constraints a geometric formulation of analytical mechanics of systems with holonomic and holonomic one sided constraints is proposed this is an extension of work by tulczyjew 1986 in which non holonomic constraints were also considered but not the one sided case the introduction of lagrange multipliers to give explicit equations of motion was not considered by tulczyjew either this allows the authors to study problems of motion of particles hitting moving walls they verify that the conditions of collision are satisfied

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

the non holonomic mechanics abstract two approaches for the study of mechanical systems with non holonomic constraints are presented d alembertian mechanics and variational vakonomic mechanics the first one is equivalent to the d alembert principle and the second comes from a variational principle in order to develop the two approaches d alembertian and vakonomic trajectories are introduced a generalized version of the classical liouville theorem for the conservation of volume is proved in the context of d alembertian mechanics a characterization for the notions of regular and singular curves is presented the flow corresponding to the regular vakonomic trajectories is hamiltonian

mechanical systems subjected to generalized non holonomic constraints we study mechanical systems subject to constraint functions that can be dependent at some points and independent at the rest such systems are modelled by means of generalized codistributions we discuss how the constraint force can transmit an impulse to the motion at the points of dependence and derive an explicit formula to obtain the post impact momentum in terms of the pre impact momentum

ideality criterion for unilateral constraints in time dependent impulsive mechanics we construct a new geometric framework based on the concepts of left and right jet bundles of a classical space time v in order to analyze the impulsive behavior of a unilateral constraint s the setup allows deep insights into how one can choose an ideality criterion for the constraint s when the hypothesis of conservation of kinetic energy is assumed we show that the conservation of kinetic energy alone univocally determines the impulsive reaction when the codimension of s is 1 and that it leaves the impulsive reaction partially undetermined when the codimension of s is greater than 1 if the codimension of s is greater than 1 we prove that an additional minimality requirement determines a physically meaningful constitutive characterization of s we show that both the newton like and the poisson like approaches to the description of the reactive impulse are equivalent in the sense that both give the same results about the ideality criterion moreover we prove that the same results hold using the cl

dynamic convexity for natural thermostatted systems abstract natural thermostatted systems are mechanical systems whose lagrangian is the difference of a kinetic and a potential energy subjected to the nonholonomic constraint of a constant kinetic energy when any two points of the configuration space are joined by a thermostatted motion we say that the system is dynamically convex a thermostatted charged particle on the plane with a constant electric field is not a dynamically convex system we prove a general sufficient condition for dynamic convexity from which whole classes of examples are easily constructed

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

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

the hegselmann krause dynamics for the continuous agent model and a regular opinion function do not always lead to consensus we present an example of a regular opinion function which as it evolves in accordance with the discrete time hegselmann krause bounded confidence dynamics always retains opinions which are separated by more than two this confirms a conjecture of blondel hendrickx and tsitsiklis

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

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

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

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

on the simultaneous presence of unilateral and kinetic constraints in time dependent impulsive mechanics the simultaneous presence of unilateral and kinetic constraints acting on a mechanical system with a finite number of degrees of freedom is framed in the geometric context of left and right jet bundles of the classical space time bundle of the system the survey gives three main cases and several subcases some of which are mathematically correct but physically meaningless the existence of at least one frame of reference for which the whole set of constraints can be at rest is the criterion selecting the physically relevant systems for these systems the conservation of kinetic energy possibly together with a standard gauss s requirement on the impulsive reaction is shown to give a well posed criterion of ideality of the constraints the application of the criterion to several examples is presented and the corresponding results are critically analyzed

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

hamiltonian theory of constrained impulsive motion this paper considers systems subject to nonholonomic constraints which are not uniform on the whole configuration manifold when the constraints change the system undergoes a transition in order to comply with the new imposed conditions building on previous work on the hamiltonian theory of impact we tackle the problem of mathematically describing the classes of transitions that can occur we propose a comprehensive formulation of the transition principle that encompasses the various impulsive regimes of hamiltonian systems our formulation is based on the partial symplectic formalism which provides a suitable framework for the dynamics of nonholonomic systems we pay special attention to mechanical systems and illustrate the results with several examples

mechanical systems subjected to impulsive constraints a geometrical description of mechanical systems subjected to impulsive non holonomic constraints is given their motions are determined by means of suitable projection operators which allow us to evaluate the instantaneous jumps of momenta due to the action of impulsive reaction forces several paradigmatic examples are investigated from this viewpoint a disk falling into a plane a rolling ball hitting a wall and the collision of two balls

dissipative systems with constraints throughout the paper m will be a c compact m dimensional connected riemannian manifold with a c riemannian metric aa4 q4 called the configuration space the dimension m of a4 is called the degree of freedom the tangent bundle 7 44 of m the velocity phase space and the cotangent bundle t m of a4 the momentum phase space the vector bundle isomorphism p tm t m defined by p u w v w is a c diffeomorphism and will be referred to as the mass operator otherwise called the legendre transformation the function k tm r defined by k v v v is called the kinetic energy of the system the constraint z is a c mapping qhzq which assigns to each qe m an n dimensional subspace c c t m the subvector bundle xm of tm defined by the vectors of all z q e m is a regular submanifold of tm and if c is the orthogonal complement of c in t m the same is true for the subvector bundle zim defined by the distribution q h zi the bundles cm elm have dimensions m n 2m n and there are canonically defined c bundle mappings p tm cm p tm 2 m defined by pv va p v v where vi v are the orthogonal projections of v on c zi by means of the mass operator one can translate em z m p

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

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

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

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

on synchronous robotic networks part ii time complexity of rendezvous and deployment algorithms this paper analyzes a number of basic coordination algorithms running on synchronous robotic networks we provide upper and lower bounds on the time complexity of the move toward average and circumcenter laws both achieving rendezvous and of the centroid law achieving deployment over a region of interest the results are derived via novel analysis methods including a set of results on the convergence rates of linear dynamical systems defined by tridiagonal toeplitz and circulant matrices

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

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

the total s energy of a multiagent system we introduce the total s energy of a multiagent system with time dependent links this provides a new analytical perspective on bidirectional agreement dynamics which we use to bound the convergence rates of dynamical systems for synchronization flocking opinion dynamics and social epistemology

nonideal unilateral constraints in impulsive mechanics a geometric approach we use the differential geometric framework of the first jet bundle of the classical space time bundle to study the impulsive behavior of a mechanical system with a finite number of degrees of freedom subject to nonideal unilateral constraints we show that this framework allows deeper insights on the concepts of nonideal constitutive characterization and of coefficient of restitution of the constraints we study the relations among newton poisson and stronge definitions of coefficient of restitution we reveal the inconsistency of the criticisms based on the energy balance of the impact for the newton definition we show the equivalence of the three definitions in the nonideal smooth case we prove the equivalence of newton and poisson ones and the insufficiency of the stronge one in the nonideal rough case we analyze the relation between coefficient of restitution and coulomb s friction coefficient in the rough case we present also several physically meaningful examples

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

geometric formulation of carnot s theorem a geometrical approach for one sided constraints is given the riemannian metric is used in order to define convenient projectors which give the post impulses in terms of the pre impulses a formulation of carnot s theorem within this geometric framework is exhibited

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

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

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

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

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

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

reduction of some classical non holonomic systems with symmetry two types of nonholonomic systems with symmetry are treated i the configuration space is a total space of a g principal bundle and the constraints are given by a connection ii the configuration space is g itself and the constraints are given by left invariant forms the proofs are based on the method of quasicoordinates in passing a derivation of the maurer cartan equations for lie groups is obtained simple examples are given to illustrate the algorithmical character of the main results

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

nonintegrability of the suslov problem in this paper we investigate the suslov problem in the case when the vector of nonholonomic constraint coincides with the third principal axis of the body and the fixed point of the body lies in the principal plane defined by the third and the first principal axes but is out of these axes we called this version of the suslov problem the generalized kozlov case and we prove that in this case a third real meromorphic first integral functionally independent together with the energy and geometrical integrals does not exist

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

geometric formulation of mechanical systems subjected to time dependent one sided constraints a geometrical setting for time dependent impulsive constraints is given this description permits us to treat in a systematic way a wide family of mechanical systems subjected to time dependent constraints like moving walls etc if the impulsive constraints remain then an affine projector can be defined giving the new initial data in terms of the old ones several examples are discussed

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

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

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

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