adaptive grid generation from harmonic maps on reimannian manifolds abstract in this paper we describe a new method for generation of solution adaptive grids based on harmonic maps on riemannian manifolds the reliability of the method is assured by an existence and uniqueness theorem for one to one maps between multidimensional multiconnected domains we formulate an adaptive riemannian metric consistent with this theorem several examples demonstrating application of the developed procedure are provided

modified signed log likelihood ratio summary the signed log likelihood ratio r for a one dimensional interest parameter can be modified as r r r log u r so that r is asymptotically standard normally distributed with error of order o n 3 2 u being interpretable as a test statistic a new more direct derivation of this result is given which leads to a more explicit expression for u and provides a better handle on the error term the relation of the tail area approximation determined by r to a generalization of the lugannani rice approximation is discussed and the applicability of r to the analysis of residual variation is indicated

local entropy averages and projections of fractal measures we show that for families of measures on euclidean space which satisfy an ergodic theoretic form of self similarity under the operation of re scaling the dimension of linear images of the measure behaves in a semi continuous way we apply this to prove the following conjecture of furstenberg let m n be integers which are not powers of the same integer and let x y be closed subsets of the unit interval which are invariant respectively under times m mod 1 and times n mod 1 then for any non zero t dim x ty min 1 dim x dim y a similar result holds for invariant measures and gives a simple proof of the rudolph johnson theorem our methods also apply to many other classes of conformal fractals and measures as another application we extend and unify results of peres shmerkin and nazarov and of moreira concerning projections of products self similar measures and gibbs measures on regular cantor sets we show that under natural irreducibility assumptions on the maps in the ifs the image measure has the maximal possible dimension under any linear projection other than the coordinate projections we also present applications to bernoulli convolutions and to the images of fractal measures under differentiable maps

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

adjusted estimates and wald statistics for the ar 1 model with constant abstract adjusted profile likelihood of cox and reid 1993 journal of the royal statistical society b 55 467 471 for the autoregressive coefficient of the ar 1 model with constant yields an estimator which can be more accurate than the maximum likelihood estimator in small samples whether the process is stationary or a random walk and in the latter case also asymptotically a related estimator is asymptotically distributed compactly and symmetrically around zero under a random walk but is not consistent in general the adjusted wald statistic can follow the asymptotic standard normal distribution much better than the non adjusted under stationarity or a drifted random walk

mesh generation using vector fields abstract a method for generating structured meshes from vector fields is described the method is based on a weighted variational principle that minimizes a least squares fit to the inverse jacobian matrix the weight is constructed from the vector fields by requiring local alignment of the mesh with the fields and by prescribing local length scales the resulting grid generator is a weighted form of the laplace grid generation equations the method is demonstrated in 2d using a subsurface flow field derived from hydrologic simulations

a framework for variational grid generation conditioning the jacobian matrix with matrix norms functionals for variational grid generation in which the jacobian matrix and matrix norms play a central role are presented all first order geometric qualities of a structured grid length area volume angle between sides aspect ratio grid alignment are determined by the jacobian matrix of the induced mapping between the logical and physical regions grids with desirable quality can be generated by requiring the jacobian matrix or the corresponding metric tensor to have certain properties functionals can be obtained by integrating over the logical or physical domain a power of the norm of a matrix derived from the jacobian matrix a judicious choice of this derived matrix results in a functional whose minima are mappings with desired features many well known grid generators fit in this framework for example the harmonic generator arises when one drives the metric tensor toward the identity the framework provides insight into what grid qualities these functionals describe three new directional control functionals are proposed and tested

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

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

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

stable and invariant adjusted directed likelihoods summary for the purpose of making inferences for a one dimensional interest parameter or constructing approximate complementary ancillaries or residuals the directed likelihood or signed square root of the likelihood ratio statistic can be adjusted so that the resulting modified directed likelihood is under ordinary repeated sampling approximately standard normal with error of o n 312 conditional on a suitable ancillary statistic and hence unconditionally in general suitable specification of the ancillary statistic may be difficult we introduce two adjusted directed likelihoods which are similar to the modified directed likelihood but do not require the specification of the ancillary statistic the error of the standard normal approximation to the distribution of these new adjusted directed likelihoods is o n 1 conditional on any reasonable ancillary statistic which is still an improvement over the unadjusted directed likelihoods

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

bnp seq bayesian nonparametric differential expression analysis of sequencing count data abstractwe perform differential expression analysis of high throughput sequencing count data under a bayesian nonparametric framework removing sophisticated ad hoc pre processing steps commonly required in existing algorithms we propose to use the gamma beta negative binomial process which takes into account different sequencing depths using sample specific negative binomial probability dispersion parameters to detect differentially expressed genes by comparing the posterior distributions of gene specific negative binomial dispersion probability parameters these model parameters are inferred by borrowing statistical strength across both the genes and samples extensive experiments on both simulated and real world rna sequencing count data show that the proposed differential expression analysis algorithms clearly outperform previously proposed ones in terms of the areas under both the receiver operating characteristic and precision recall curves supplementary materials for this article are avai

improved confidence intervals for the difference between binomial proportions based on paired data existing methods for setting confidence intervals for the difference theta between binomial proportions based on paired data perform inadequately the asymptotic method can produce limits outside the range of validity the exact conditional method can yield an interval which is effectively only one sided both these methods also have poor coverage properties better methods are described based on the profile likelihood obtained by conditionally maximizing the proportion of discordant pairs a refinement methods 5 and 6 which aligns 1 alpha with an aggregate of tail areas produces appropriate coverage properties a computationally simpler method based on the score interval for the single proportion also performs well method 10

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

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

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

a conditional likelihood approach to residual maximum likelihood estimation in generalized linear models summary residual maximum likelihood reml estimation is often preferred to maximum likelihood estimation as a method of estimating covariance parameters in linear models because it takes account of the loss of degrees of freedom in estimating the mean and produces unbiased estimating equations for the variance parameters in this paper it is shown that reml has an exact conditional likelihood interpretation where the conditioning is on an appropriate sufficient statistic to remove dependence on the nuisance parameters this interpretation clarifies the motivation for reml and generalizes directly to non normal models in which there is a low dimensional sufficient statistic for the fitted values the conditional likelihood is shown to be well defined and to satisfy the properties of a likelihood function even though this is not generally true when conditioning on statistics which depend on parameters of interest using the conditional likelihood representation the concept of reml is extended to generalized linear models with varying dispersion and canonical link explicit calculation of the conditional likelihood is given for the one way lay out a saddlepoint approximation for the conditional likelihood is also derived

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

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

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

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

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

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

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

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

variational mesh adaptation methods for axisymmetrical problems we study variational mesh adaptation for axially symmetric solutions to two dimensional problems the study is focused on the relationship between the mesh density distribution and the monitor function and is carried out for a traditional functional that includes several widely used variational methods as special cases and a recently proposed functional that allows for a weighting between mesh isotropy or regularity and global equidistribution of the monitor function the main results are stated in theorems ref thm4 1 and ref thm4 2 for axially symmetric problems it is natural to choose axially symmetric mesh adaptation to this end it is reasonable to use the monitor function in the form g lambda 1 r mbox boldmath e r mbox boldmath e r t lambda 2 r mbox boldmath e theta mbox boldmath e theta t where mbox boldmath e r and mbox boldmath e theta are the radial and angular unit vectors it is shown that when higher mesh concentration at the origin is desired a choice of lambda 1 and lambda 2 satisfying lambda 1 0 0 by choosing lambda 1 and lambda 2 in contrast numerical results show that the new functional provides better control of the mesh concentration through the monitor function two dimensional numerical results are presented to support the analysis

interval estimation of the over dispersion parameter in the analysis of one way layout of count data the over dispersion parameter is an important and versatile measure in the analysis of one way layout of count data in biological studies for example it is commonly used as an inverse measure of aggregation in biological count data its estimation from finite data sets is a recognized challenge many simulation studies have examined the bias and efficiency of different estimators of the over dispersion parameter for finite data sets see for example clark and perry biometrics 1989 45 309 316 and piegorsch biometrics 1990 46 863 867 but little attention has been paid to the accuracy of the confidence intervals cis of it in this paper we first derive asymptotic procedures for the construction of confidence limits for the over dispersion parameter using four estimators that are specified by only the first two moments of the counts we also obtain closed form asymptotic variance formulae for these four estimators in addition we consider the asymptotic ci based on the maximum likelihood ml estimator using the negative binomial model it appears from the simulation results that the asymptotic cis based on these five estimators have coverage below the nominal coverage probability to remedy this we also study the properties of the asymptotic cis based on the restricted estimates of ml extended quasi likelihood and double extended quasi likelihood by eliminating the nuisance parameter effect using their adjusted profile likelihood and quasi likelihoods it is shown that these cis outperform the competitors by providing coverage levels close to nominal over a wide range of parameter combinations two examples to biological count data are presented copyright 2010 john wiley sons ltd

a variationally consistent mesh adaptation method for triangular elements in explicit lagrangian dynamics in this paper a variational formulation for mesh adaptation procedures involving local mesh changes for triangular meshes is presented such local adaptive changes are very well suited for explicit methods as they do not involve significant computational expense they also greatly simplify the projection of field variables from the old to the new meshes crucially the variational nature of the formulation used to derive the equilibrium equations at steps where adaptation takes place ensures that conservation of linear and angular momentum is obtained int j numer meth engng 2000 49 1295 1325 several examples in 2 d showing the application of the proposed adaptive algorithms are used to demonstrate the validity of the methodology proposed copyright 2009 john wiley sons ltd

practical aspects of formulation and solution of moving mesh partial differential equations moving mesh partial differential equations mmpdes are used in the mmpde moving mesh method to generate adaptive moving meshes for the numerical solution of time dependent problems how mmpdes are formulated and solved is crucial to the efficiency and robustness of the method in this paper several practical aspects of formulating and solving mmpdes are studied they include spatial balance scaling invariance effective control of mesh concentration bounds on time steps multiple sub steps and two level mesh movement numerical results are also given c 2001 academic press

a simple predictive density function abstract an easily computable predictive density is considered although involving a plain maximum likelihood technique it coincides with the posterior predictive density with a noninformative prior for standard situations achievements are comparable with classical inference

similarity between the mandelbrot set and julia sets the mandelbrot setm is self similar about any misiurewicz pointc in the sense that if we examine a neighborhood ofc inm with a very powerful microscope and then increase the magnification by a carefully chosen factor the picture will be unchanged except for a rotation the corresponding julia setj c is also self similar in the same sense with the same magnification factor moreover the two setsm andj c are similar in the sense that if we use a very powerful microscope to look atm andj c both focused atc the structures we see look like very much the same

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

generalized linear models with varying dispersion summary generalized linear models are further generalized to include a linear predictor for the dispersion as well as for the mean it is shown how the convenient structure of generalized linear models can be carried over to this more general setting by considering the mean and dispersion structure separately mean and dispersion submodels are formulated for this the dependent variable for the dispersion submodel being the deviance components of the mean submodel the fact that both submodels are essentially generalized linear models themselves is used to derive simple expressions for the likelihood equations and for asymptotic tests estimation algorithms are proposed which have good convergence properties the results apply mainly to the normal inverse gaussian and gamma distributions but can be extended to discrete distributions by using quasi likelihoods the methods developed are applied to a well known data set

estimation of spatial autoregressive panel data models with fixed effects this paper establishes asymptotic properties of quasi maximum likelihood estimators for sar panel data models with fixed effects and sar disturbances a direct approach is to estimate all the parameters including the fixed effects because of the incidental parameter problem some parameter estimators may be inconsistent or their distributions are not properly centered we propose an alternative estimation method based on transformation which yields consistent estimators with properly centered distributions for the model with individual effects only the direct approach does not yield a consistent estimator of the variance parameter unless t is large but the estimators for other common parameters are the same as those of the transformation approach we also consider the estimation of the model with both individual and time effects

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

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

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

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

improving the accuracy of likelihood based inference in meta analysis and meta regression random effects models are frequently used to synthesize information from different studies in meta analysis while likelihood based inference is attractive both in terms of limiting properties and of implementation its application in random effects meta analysis may result in misleading conclusions especially when the number of studies is small to moderate the current paper shows how methodology that reduces the asymptotic bias of the maximum likelihood estimator of the variance component can also substantially improve inference about the mean effect size the results are derived for the more general framework of random effects meta regression which allows the mean effect size to vary with study specific covariates

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

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

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

improved estimation in cumulative link models for the estimation of cumulative link models for ordinal data the bias reducing adjusted score equations of firth in 1993 are obtained whose solution ensures an estimator with smaller asymptotic bias than the maximum likelihood estimator their form suggests a parameter dependent adjustment of the multinomial counts which in turn suggests the solution of the adjusted score equations through iterated maximum likelihood fits on adjusted counts greatly facilitating implementation like the maximum likelihood estimator the reduced bias estimator is found to respect the invariance properties that make cumulative link models a good choice for the analysis of categorical data its additional finiteness and optimal frequentist properties along with the adequate behaviour of related asymptotic inferential procedures make the reduced bias estimator attractive as a default choice for practical applications furthermore the estimator proposed enjoys certain shrinkage properties that are defensible from an experimental point of view relating to the nature of ordinal data

a maximum likelihood method for the incidental parameter problem this paper uses the invariance principle to solve the incidental parameter problem of 35 we seek group actions that pre serve the structural parameter and yield a maximal invariant in the parameter space with flxed dimension m estimation from the likeli hood of the maximal invariant statistic yields the maximum invariant likelihood estimator mile consistency of mile for cases in which the likelihood of the maximal invariant is the product of marginal likelihoods is straightforward we illustrate this result with a sta tionary autoregressive model with flxed efiects and an agent speciflc monotonic transformation model asymptotic properties of mile when the likelihood of the maxi mal invariant does not factorize remain an open question we are able to provide consistency asymptotic normality and e ciency results of mile when invariance yields wishart distributions two examples are an instrumental variable iv model and a dynamic panel data model with flxed efiects

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

on confidence intervals associated with the usual and adjusted likelihoods in the presence of nuisance parameters we derive an explicit higher order asymptotic formula to compare the expected lengths of confidence intervals given by likelihood ratio statistics arising from the usual profile likelihood and various adjustments thereof highest posterior density regions with approximate frequentist validity are also included in the study

variational mesh adaptation isotropy and equidistribution we present a new approach for developing more robust and error oriented mesh adaptation methods specifically assuming that a regular in cell shape and uniform in cell size computational mesh is used as is commonly done in computation we develop a criterion for mesh adaptation based on an error function whose definition is motivated by the analysis of function variation and local error behavior for linear interpolation the criterion is then decomposed into two aspects the isotropy or conformity and uniformity or equidistribution requirements each of which can be easier to deal with the functionals that satisfy these conditions approximately are constructed using discrete and continuous inequalities a new functional is finally formulated by combining the functionals corresponding to the isotropy and uniformity requirements the features of the functional are analyzed and demonstrated by numerical results in particular unlike the existing mesh adaptation functionals the new functional has clear geometric meanings of minimization a mesh that has the desired properties of isotropy and equidistribution can be obtained by properly choosing the values of two parameters the analysis presented in this article also provides a better understanding of the increasingly popular method of harmonic mapping in two dimensions

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

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

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

an iterative grid redistribution method for singular problems in multiple dimensions we introduce an iterative grid redistribution method based on the variational approach the iterative procedure enables us to gain more precise control of the grid distribution near the regions of large solution variations the method is particularly effective for solving partial differential equations with singular solutions e g blowup solutions our method requires little prior information of the singular solutions and can handle multiple singularities the method is successfully applied to the nonlinear schrodinger equation and the keller segal equations where solutions with multiple blowup points can be solved up to times very close to the blowup time

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

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

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

information bias and adjusted profile likelihoods summary the bias and information bias of the ordinary profile score statistic are both typically of order 0 1 several additive adjustments to the profile score statistic that reduce its bias to order o n have been proposed in certain situations the information bias of these adjusted profile score statistics is also reduced to order o n q specifically the modified profile likelihood of barndorff nielsen yields an adjusted profile score statistic having both reduced bias and reduced information bias in general however a bias reducing adjustment to the profile score statistic will not automatically reduce the order of the information bias as well in this paper an analytical formula is obtained for the information bias of a general bias adjusted profile score statistic this formula is used to compare bias reducing adjustments to the profile score statistic as well as to construct further adjustments that reduce the information bias to o n 1 several examples are presented to illustrate use of the formula for information bias in particular the information bias formula may be utilized in a criterion for choosing between orthogonal parameterizations for the conditional profile likelihood of cox and reid

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

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

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

variational mesh adaptation ii error estimates and monitor functions the key to the success of a variational mesh adaptation method is to define a proper monitor function which controls mesh adaptation in this paper we study the choice of the monitor function for the variational adaptive mesh method developed in the previous work j comput phys 174 2001 924 two types of monitor functions scalar matrix and non scalar matrix ones are defined based on asymptotic estimates of interpolation error obtained using the interpolation theory of finite element methods the choice of the adaptation intensity parameter is also discussed for each of these monitor functions asymptotic bounds on interpolation error are obtained for adaptive meshes that satisfy the regularity and equidistribution conditions two dimensional numerical results are given to verify the theoretical findings

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

on the equivalence of prospective and retrospective likelihood methods in case control studies summary we present new approaches to analyzing case control studies using prospective likelihood methods in the classical framework we extend the equality of the profile likelihoods to the barndorff nielsen modified profile likelihoods for prospective and retrospective models this enables simple and accurate approximate conditional inference for stratified case control studies of moderate stratum size in the bayesian framework we provide sufficient conditions on priors for the prospective model parameters to yield a prospective marginal posterior density equal to its retrospective counterpart our results extend the prospective retrospective equivalence in the bayesian paradigm with a more general class of priors than has previously been investigated

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

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

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

a study of monitor functions for two dimensional adaptive mesh generation in this paper we study the problem of two dimensional adaptive mesh generation using a variational approach and specifically the effect that the monitor function has on the resulting mesh behavior the basic theoretical tools employed are green s function for elliptic problems and the eigendecomposition of symmetric positive definite matrices based upon this study a general strategy is suggested for how to choose the monitor function and numerical results are presented for illustrative purposes the three dimensional case is also briefly discussed it is noted that the strategy used here can be applied to other elliptic mesh generation techniques as well

the scenery flow for hyperbolic julia sets we define the scenery flow space at a point z in the julia set j of a hyperbolic rational map t c c with degree at least 2 and more generally for t a conformal mixing repellor we prove that for hyperbolic rational maps except for a few exceptional cases listed below the scenery flow is ergodic we also prove ergodicity for almost all conformal mixing repellors here the statement is that the scenery flow is ergodic for the repellors which are not linear nor contained in a finite union of real analytic curves and furthermore that for the collection of such maps based on a fixed open set u the ergodic cases form a dense open subset of that collection scenery flow ergodicity implies that one generates the same scenery flow by zooming down towards almost every z with respect to the hausdorff measure hd where d is the dimension of j and that the flow has a unique measure of maximal entropy for all conformal mixing repellors the flow is loosely bernoulli and has topological entropy at most d moreover the flow at almost every point is the same up to a rotation and so as a corollary one has an analogue of the lebesgue density theorem for the fractal set giving a different proof of a theorem of falconer

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

multiparameter testing in exponential models third order approximations from likelihood summary for composite testing of several canonical components in an exponential model a conical or directional test provides a one dimensional measure of departure we approximate the distribution of this test statistic to third order by numerical inversion of a profile likelihood the method extends the lugannani rice type formula to kernels other than the normal

marginal likelihood and unit roots we develop new tests for the hypothesis of unit roots that are based on the marginal likelihood of the general linear model the marginal likelihood allows the incorporation of invariance arguments in the likelihood function it turns out that marginal likelihood tests for unit roots appear to be more powerful than other unit root tests for some basic models power functions almost coincide with the power envelopes even in small samples general correlation structures can be incorporated either by standard likelihood procedures or by adjustments of the test statistics on the basis of asymptotic distributions

jacobian weighted elliptic grid generation variational grid generation techniques are used to derive and analyze a weighted elliptic grid generator that controls the jacobian of the underlying transformation in a least squares sense the euler lagrange equations for the area and volume generators are weighted forms of the well known laplace generator weights are restricted to the class of p matrices to help achieve global invertibility of the map connecting the weights to the jacobian of the map results in an intuitive means of controlling grid spacing area orthogonality and grid line directions examples are given on the unit square to demonstrate point attraction local refinement directional alignment and adaption to a shock

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