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

bayesian 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

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

practical use of higher order asymptotics for multiparameter exponential families recently developed asymptotics based on saddlepoint methods provide important practical methods for multiparameter exponential families especially in generalized linear models the aim here is to clarify and explore these attention is restricted to tests and confidence intervals regarding a single parametric function which can be represented as a natural parameter of a full rank exponential family excellent approximations to exact conditional inferences are often available in terms of simple adjustments to the signed square root of the likelihood ratio statistic

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

adjustments to profile likelihood summary conditional and marginal likelihoods constructed from parameter dependent functions of the data have an additional parameter dependence that changes with the choice of supporting metric for the corresponding densities we consider constructing marginal and conditional likelihoods by using densities expressed in terms of an intrinsic choice of support measure based on either euclidean metric or the constant information metric barndorff nielsen s 1983 approximation to the distribution of the maximum likelihood estimate is used to approximate these conditional and marginal likelihoods the resulting likelihoods are closely related to the modified profile likelihood of barndorff nielsen 1983 and the approximate conditional likelihood of cox reid 1987 the normal circle problem is examined in some detail

studentization and deriving accurate p values the original studentization was the conversion of a sample mean departure into the familiar t statistic plus the derivation of the corresponding student distribution function the observed value of the distribution function is the observed p value as presented in an elemental form we examine this process in a broadly general context a null statistical model is available together with observed data a statistic t y has been proposed as a plausible measure of the location of the data relative to what is expected under the null a modified statistic say t y is developed that is ancillary the corresponding distribution function is determined exactly or approximately and the observed value of the distribution function is the p value or percentage position of the data with respect to the model such p values have had extensive coverage in the recent bayesian literature with many variations and some preference for two versions labelled pppost and pcpred the bootstrap method also directly addresses this studentization process we use recent likelihood theory that gives a third order factorization of a regular statistical model into a marginal density for a full dimensional ancillary and a conditional density for the maximum likelihood variable the full dimensional ancillary is shown to lead to an explicit determination of the studentized version t y together with a highly accurate approximation to its distribution function the observed value of the distribution function is the p value and is available numerically by direct calculation or by markov chain monte carlo or by other simulations in this paper for any given initial or trial test statistic proposed as a location indicator for a data point we develop an ancillary based p value designated panc a special version of the bayesian pcpred and a bootstrap based p value designated pbs we then show under moderate regularity that these are equivalent to the third order and have uniqueness as a determination of the statistical location of the data point as of course derived from the initial location measure we also show that these p values have a uniform distribution to third order as based on calculations in the moderate deviations region for implementation the bayesian and likelihood procedures would perhaps require the same numerical computations while the bootstrap would require a magnitude more in computation and would perhaps not be accessible examples are given to indicate the ease and exibility of the approach

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

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

single cell discretizations of order two and four for biharmonic problems abstract new difference formulas are derived for solving the biharmonic problem in two dimensions over a rectangular domain these methods use only the nine grid points of a single mesh cell and do not require fictitious points in order to approximate the boundary conditions derivatives of the solution are obtained as a by product of the methods second order formulas are derived for both the first and second biharmonic problems in numerical experiments the new second order formulas compare favourably with the standard second order methods extensions to fourth order formulas are given the method of deriving these formulas can be used to derive similar formulas for arbitrarily shaped regions

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

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

integrated likelihood functions for non bayesian inference consider a model with parameter 0 where is the parameter of interest and let l denote the likelihood function one approach to likelihood inference for is to use an integrated likelihood function in which is eliminated from l by integrating with respect to a density function the goal of this paper is to consider the problem of selecting so that the resulting integrated likelihood function is useful for non bayesian likelihood inference the desirable properties of an integrated likelihood function are analyzed and these suggest that should be chosen by finding a nuisance parameter that is unrelated to and then taking the prior density for to be independent of such an unrelated parameter is constructed and the resulting integrated likelihood is shown to be closely related to the modified profile likelihood

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

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

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

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

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

unconditionally stable finite difference scheme and iterative solution of 2d microscale heat transport equation abstract a two dimensional time dependent heat transport equation at the microscale is derived a second order finite difference scheme in both time and space is introduced and the unconditional stability of the finite difference scheme is proved a computational procedure is designed to solve the discretized linear system at each time step by using a preconditioned conjugate gradient method numerical results are presented to validate the accuracy of the finite difference scheme and the efficiency of the proposed computational procedure

high order difference methods for heat equations in polar cylindrical coordinates abstract fourth order difference methods for the solution of poisson equations in cylindrical polar coordinates are proposed the same technique is then applied to obtain o k 2 h 4 two level unconditionally stable adi methods for the solution of the heat equation in two dimensional polar coordinates and three dimensional cylindrical coordinates numerical examples given here show that the methods developed here retain their order and accuracy everywhere including the region in the vicinity of the singularity r 0

an approximation to the modified profile likelihood function an approximation to the modified profile likelihood function is proposed this approximation is invariant under interest respecting reparameterisations it agrees with the modified profile likelihood function to order o n 1 in the moderate deviation sense and to order o n 1 2 in the large deviation sense and it is stable in the sense of conditional inference for the case of a curved exponential family model this approximation agrees with the one proposed by barndorff nielsen 1995 a general expression for the approximate modified profile likelihood function is given for nonlinear regression models and for generalised linear models with known dispersion parameter

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

likelihood ratio statistics based on an integrated likelihood an integrated likelihood depends only on the parameter of interest and the data so it can be used as a standard likelihood function for likelihood based inference in this paper the higher order asymptotic properties of the signed integrated likelihood ratio statistic for a scalar parameter of interest are considered these results are used to construct a modified integrated likelihood ratio statistic and to suggest a class of prior densities to use in forming the integrated likelihood the properties of the integrated likelihood ratio statistic are compared to those of the standard likelihood ratio statistic several examples show that the integrated likelihood ratio statistic can be a useful alternative to the standard likelihood ratio statistic

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

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

a 2 approximation algorithm for genome rearrangements by reversals and transpositions recently a new approach to analyze genomes evolving which is based on comparision of gene orders versus traditional comparision of dna sequences was proposed sankoff et al 1992 the approach is based on the global rearrangements e g inversions and transpositions of fragments analysis of genomes evolving by inversions and transpositions leads to a combinatorial problem of sorting by reversals and transpositions i e sorting of a permutation using reversals and transpositions of arbitrary fragments we study sorting of signed permutations by reversals and transpositions a problem which adequately models genome rearrangements as the genes in dna are oriented we establish a lower bound and give a 2 approximation algorithm for the problem

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

high accuracy difference schemes for the system of two space nonlinear parabolic differential equations with mixed derivatives and variable coefficients abstract in this article two level compact implicit difference methods of o k2 kh2 h4 using 9 spatial grid points are proposed for the numerical solution of the system of two dimensional nonlinear parabolic equations with variable coefficients subject to the dirichlet boundary conditions where k 0 and h 0 are step lengths in time and space directions respectively the proposed difference method for scalar equation is applied for the solution of the heat conduction equation in polar coordinates to obtain the two level unconditionally stable adi method of o k2 h4 the method having two variables also has been successfully applied on two dimensional unsteady navier stokes model equations in polar coordinates some numerical examples are presented to demonstrate the accuracy of the implementation

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

a new fourth order discretization for singularly perturbed two dimensional non linear elliptic boundary value problems abstract in this article we derive a new difference method of o h4 so called arithmetic average discretization for the solution of two dimensional non linear singularly perturbed elliptic partial differential equation of the form e uxx uyy f x y u ux uy 0 0 is a small parameter we also derive new methods of o h4 for the estimates of u n which are quite often of interest in many physical problems in all cases we require only 9 grid points and a single computational cell the main advantage of the proposed methods is that the methods are directly applicable to singular problems we do not require any special technique or modification to solve singular problems numerical results are provided to demonstrate the usefulness of the methods discussed

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

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

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

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

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

reproducing kernel method for numerical simulation of downhole temperature distribution abstract this paper research downhole temperature distribution in oil production and water injection using reproducing kernel hilbert space method rkhsm for the first time the aim of this paper is that using the highly accurate rkhsm can solve downhole temperature problems effectively according to 2 d mathematical models of downhole temperature distribution the analytical solution was given in a series expansion form and the approximate solution was expressed by n term summation of reproducing kernel functions which initial and boundary conditions were selected properly numerical results of downhole temperature distribution with multiple pay zones in which different radial distance and different injection production conditions such as injection rate injection temperature injection time oil layer thickness were carried out by mathematical 7 0 and numerical results correspond to general knowledge and show that use rkhsm to research downhole temperature distribution is feasible and effective

a new two level implicit discretization of o k 2 kh 2 h 4 for the solution of singularly perturbed two space dimensional non linear parabolic equations we propose a new two level implicit difference method of o k2 kh2 h4 for the solution of singularly perturbed non linear parabolic differential equation e uxx uyy f x y t u ux u y ut 0 0 subject to appropriate initial and dirichlet boundary conditions where k 0 and h 0 are grid sizes in time and space directions respectively and e 0 is a small parameter we also develop new methods of o kh2 h4 for the estimates of u x and u y in all cases we use 9 spatial grid points and a single computational cell the proposed methods are directly applicable to singular problems we do not require any special scheme to solve singular problems we also discuss alternating direction implicit adi method for solving diffusion equation in polar cylindrical coordinates this method permits multiple use of the one dimensional tri diagonal algorithm with a considerable saving in computing time and produces a very efficient solver it is shown that the adi method is unconditionally stable numerical experiments are conducted to test the high accuracy of the proposed methods and compared with the exact solutions

reversal and transposition medians in determining phylogenetic trees using gene order information medians provide a powerful alternative to pairwise distances on the other hand both breakpoint and reversal medians are np hard to compute and the use of medians has been limited to relatively closely related genomes in this paper we show that in spite of the greater non uniqueness of reversal medians compared to breakpoint medians medians of moderately distant genomes are often widely spread this means that regardless of which approximation algorithms one may devise for computing reversal medians the genomes need to be closely related for phylogenetic tree computations to be successful to show this we use results on transposition medians which behave similarly and also support our claims with simulations and a real data example with widely spread medians

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 simple general formula for tail probabilities for frequentist and bayesian inference summary we describe a simple general formula for approximating the p value for testing a scalar parameter in the presence of nuisance parameters the formula covers both frequentist and bayesian contexts and does not require explicit nuisance parameterisation implementation is discussed in terms of computer algebra packages examples are given and the relationship to barndorff nielsen s approximation is discussed

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

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

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

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

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

frequentist and bayesian bartlett correction of test statistics based on adjusted profile likelihoods summary for inference about a multidimensional parameter b in the presence of nuisance param

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

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

improved likelihood inference for discrete data reference stat article 2006 001doi 10 1111 j 1467 9868 2006 00548 xview record in web of science record created on 2006 04 21 modified on 2017 05 12

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

the transposition median problem is np complete during recent years the genomes of more and more species have been sequenced providing data for phylogenetic reconstruction based on genome rearrangement measures where the most important distance measures are the reversal distance and the transposition distance the two main tasks in all phylogenetic reconstruction algorithms are to calculate pairwise distances and to solve the median of three problem while the reversal distance problem can be solved in linear time the reversal median problem has been proven to be np complete the status of the transposition distance problem is still open but it is conjectured to be more difficult than the reversal problem this in turn also suggests that the transposition median problem is np complete however this conjecture could not yet be proven we have now succeeded in giving a non trivial proof for the np completeness of the transposition median problem

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

regression analysis nonlinear or nonnormal simple and accurate p values from likelihood analysis we develop simple approximations for the p values to use with regression models having linear or nonlinear parameter structure and normal or nonnormal error distribution computer iteration then gives confidence intervals both frequentist and bayesian versions are given the approximations are derived from recent developments in likelihood analysis and have third order accuracy also for very small and medium sized samples the accuracy can typically be high the likelihood basis of the procedure seems to provide the grounds for this general accuracy examples are discussed and simulations record the distributional accuracy

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 new high accuracy two level implicit off step discretization for the system of two space dimensional quasi linear parabolic partial differential equations abstract in this article we propose a two level compact implicit off step discretization for the solution of the system of 2d quasi linear parabolic partial differential equations subject to suitable initial and boundary conditions new methods for the estimates of first order space derivatives of the solution are also derived these methods are fourth order accurate in space and second order accurate in time and involve only nine spatial grid points of a single compact cell we further develop the alternating direction implicit adi scheme for a general linear parabolic equation which is shown to be unconditionally stable for the heat equation in polar coordinates the proposed methods are directly applicable to singular problems without the need of any special technique which is the main advantage of this work the method is effectively applied to the time dependent navier stokes model equations in polar coordinates numerical examples are provided to illustrate the accuracy of the methods

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

a faster and simpler algorithm for sorting signed permutations by reversals we give a quadratic time algorithm for finding the minimum number of reversals needed to sort a signed permutation our algorithm is faster than the previous algorithm of hannenhalli and pevzner and its faster implementation by berman and hannenhalli the algorithm is conceptually simple and does not require special data structures our study also considerably simplifies the combinatorial structures used by the analysis

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

1 ɛ approximation of sorting by reversals and transpositions gu et al gave a 2 approximation for computing the minimal number of inversions and transpositions needed to sort a permutation there is evidence that from the point of view of computational molecular biology a more adequate objective function is obtained if transpositions are given double weight we present a 1 e approximation for this problem based on the exact algorithm of hannenhalli and pevzner for sorting by reversals only

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

default priors for bayesian and frequentist inference summary we investigate the choice of default priors for use with likelihood for bayesian and frequentist inference such a prior is a density or relative density that weights an observed likelihood function leading to the elimination of parameters that are not of interest and then a density type assessment for a parameter of interest for independent responses from a continuous model we develop a prior for the full parameter that is closely linked to the original bayes approach and provides an extension of the right invariant measure to general contexts we then develop a modified prior that is targeted on a component parameter of interest and by targeting avoids the marginalization paradoxes of dawid and co workers this modifies jeffreys s prior and provides extensions to the development of welch and peers these two approaches are combined to explore priors for a vector parameter of interest in the presence of a vector nuisance parameter examples are given to illustrate the computation of the priors