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

downlink throughput maximization for ofdma systems with feedback channel capacity constraints in this correspondence we study the downlink throughput maximization for orthogonal frequency division multiple access ofdma systems in the presence of feedback channel capacity constraints we establish an information theoretic lower bound on the capacity of feedback channel and build the corresponding test channel that achieves this lower bound based on the derived test channel we formulate two optimization problems that maximize the downlink throughput with quantized channel state information csi i one problem where throughput is defined with the ergodic throughput and ii the other problem with the outage throughput from which we can see the performance limit for given limit feedback channels we solve the throughput maximization problem through an iterative approach which achieves the optimal ergodic throughput and the near optimal outage throughput numerical results show that the downlink throughput with a limited feedback of csi can be close to that with perfect csi by exploiting correlation properties of downlink csi for quantization

two dimensional stationary schrodinger equation via the dressing method new exactly solvable potentials wave functions and their physical interpretation the classes of exactly solvable multiline soliton potentials and corresponding wave functions of two dimensional stationary schrodinger equation via dressing method are constructed and their physical interpretation is discussed

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

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

approximately bayesian inference abstract consider statistical inference about a scalar parameter and suppose that information about that parameter is to be summarized by a system of interval estimates it is well known that methods of interval estimation that do not correspond to bayesian inference with respect to some prior distribution have some logical difficulties this article proposes a measure of how close a system of interval estimates is to being of bayesian form using this measure several non bayesian methods of interval estimation are analyzed these results are illustrated on several examples

reml estimation asymptotic behavior and related topics the restricted maximum likelihood reml estimates of dispersion parameters variance components in a general non normal mixed model are defined as solutions of the reml equations in this paper we show the reml estimates are consistent if the model is asymptotically identifiable and infinitely informative under the location invariant class and are asymptotically normal a n if in addition the model is asymptotically nondegenerate the result does not require normality or boundedness of the rank p of design matrix of fixed effects moreover we give a necessary and sufficient condition for asymptotic normality of gaussian maximum likelihood estimates mle in non normal cases as an application we show for all unconfounded balanced mixed models of the analysis of variance the reml anova estimates are consistent and are also a n provided the models are nondegenerate the mle are consistent a n if and only if certain constraints on p are satisfied

approximations for densities of sufficient estimators abstract a simple method of obtaining asymptotic expansions for the densities of sufficient estimators is described it is an extension of the one developed by o barndorff nielsen and d r cox 1979 for exponential families a series expansion in powers of n 1 is derived of which the first term has an error of order n 1 which can effectively be reduced to n 3 2 by renormalization the results obtained are similar to those given by h e daniels s 1954 saddlepoint method but the derivations are simpler a brief treatment of approximations to conditional densities is given theorems are proved which extend the validity of the multivariate edgeworth expansion to parametric families of densities of statistics which need not be standardized sums of independent and identically distributed vectors these extensions permit the treatment of problems arising in time series analysis the technique is used by j durbin 1980 to obtain approximations to the densities of partial serial correlation coefficients

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

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

on the structure of transformation models in this paper we characterize transformation models by means of the functional form of the densities we discuss sufficiency of the pair t where t is an equivariant estimator and v is a maximal invariant furthermore we introduce and discuss the algebraic concept of structural sufficiency this gives rise to an example of a transformation model where t a is nonsufficient 1 introduction in the analysis of statistical models it is sometimes convenient to make use of invariance properties of the model in question for instance the invariance principle see lehmann 20 or hall wijsman and ghosh 16 is a widely accepted and frequently used statistical tool closely related to this concept is the notion of transformation models let e be a sample space e a parameter set and g a group acting on e and e in our setup a transformation model is a family of probability measures p 9 with the property

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

effects of the reference set on frequentist inferences summary we employ second order likelihood asymptotics to investigate how ideal frequentist inferences depend on the probability model for the data through more than the likelihood function referring to this as the effect of the reference set there are two aspects of higher order corrections to first order likelihood methods namely i that involving effects of fitting nuisance parameters and leading to the modified profile likelihood and ii another part pertaining to limitation in adjusted information generally each of these involves a first order adjustment depending on the reference set however we show that for some important settings likelihood irrelevant model specifications have a second order effect on both of these adjustments this result includes specification of the censoring model for survival data on the other hand for sequential experiments the likelihood irrelevant specification of the stopping rule has a second order effect on adjustment i but a first order effect on adjustment ii these matters raise the issue of what are ideal frequentist inferences since consideration of exact frequentist inferences will not suffice we indicate that to second order ideal frequentist inferences may be based on the distribution of the ordinary likelihood ratio statistic without commonly considered adjustments thereto

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

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

weighted likelihood estimation of ability in item response theory applications of item response theory which depend upon its parameter invariance property require that parameter estimates be unbiased a new method weighted likelihood estimation wle is derived and proved to be less biased than maximum likelihood estimation mle with the same asymptotic variance and normal distribution wle removes the first order bias term from mle two monte carlo studies compare wle with mle and bayesian modal estimation bme of ability in conventional tests and tailored tests assuming the item parameters are known constants the monte carlo studies favor wle over mle and bme on several criteria over a wide range of the ability scale

general saddlepoint approximations with applications to l statistics abstract saddlepoint approximations are extended to general statistics the technique is applied to derive approximations to the density of linear combinations of order statistics including trimmed means a comparison with exact results shows the accuracy of these approximations even in very small sample sizes

groups acting on gaussian graphical models gaussian graphical models have become a well recognized tool for the analysis of conditional independencies within a set of continuous random variables from an inferential point of view it is important to realize that they are composite exponential transformation families we reveal this structure by explicitly describing for any undirected graph the maximal matrix group acting on the space of concentration matrices in the model the continuous part of this group is captured by a poset naturally associated to the graph while automorphisms of the graph account for the discrete part of the group we compute the dimension of the space of orbits of this group on concentration matrices in terms of the combinatorics of the graph and for dimension zero we recover the characterization by letac and massam of models that are transformation families furthermore we describe the maximal invariant of this group on the sample space and we give a sharp lower bound on the sample size needed for the existence of equivariant estimators of the concentration matrix finally we address the issue of robustness of these estimators by computing upper bounds on finite sample breakdown points keywords gaussian graphical models covariance matrix concentration matrix robust estimator breakdown point equivariant estimator transformation families

median bias reduction of maximum likelihood estimates for regular parametric problems we show how median centring of the maximum likelihood estimate can be achieved by a simple modification of the score equation for a scalar parameter of interest the estimator is equivariant under interest respecting reparameterizations and is third order median unbiased with a vector parameter of interest componentwise equivariance and third order median centring are obtained like the implicit method of firth 1993 for bias reduction the new method does not require finiteness of the maximum likelihood estimate and is effective in preventing infinite estimates simulation results for continuous and discrete models including binary and beta regression confirm that the method succeeds in achieving componentwise median centring and in solving the boundary estimate problem while keeping comparable dispersion and the same approximate distribution as its main competitors

approximate predictive pivots and densities summary this paper suggests two predictive likelihoods that can be applied in almost any parametric model setting the first can sometimes be interpreted as an approximate predictive pivot barnard 1986 while the second is often an approximation to a bayesian predictive density with a flat prior the issue of calibrating various predictive likelihoods in terms of long run predictive coverage is also discussed and a specific criterion by which these likelihoods can be compared is proposed

a symbolic algorithm for studying adjustments to the profile likelihood summary a symbolic algorithm for the derivation of the asymptotic expansion of the maximum likelihood estimate was first introduced by andrews stafford 1993 this algorithm is quite general and can be modified to give asymptotic expansions for the maximum likelihood estimate based on either the conditional modified or adjusted profile likelihoods expansions are obtained quickly with no chance for human clerical error as a result these three adjustments can be easily studied and some new statistical results are obtained bartlett corrections of each adjusted likelihood and asymptotic expansions for the first four moments of the maximum likelihood estimate based on the conditional and adjusted profile likelihoods are derived

parameter estimation and inference in the linear mixed model the paper reviews the linear mixed model with a focus on parameter estimation and inference parameter estimation for the different components of the model are reviewed with an emphasis on variance parameter estimation inferential procedures for the fixed effects random effects or a combination of both fixed and random effects are also 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

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

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

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

conditional inferential models combining information for prior free probabilistic inference summary the inferential model im framework provides valid prior free probabilistic inference by focusing on predicting unobserved auxiliary variables but efficient im based inference can be challenging when the auxiliary variable is of higher dimension than the parameter here we show that features of the auxiliary variable are often fully observed and in such cases a simultaneous dimension reduction and information aggregation can be achieved by conditioning this proposed conditioning strategy leads to efficient im inference and casts new light on fisher s notions of sufficiency conditioning and also bayesian inference a differential equation driven selection of a conditional association is developed and validity of the conditional im is proved under some conditions for problems that do not admit a conditional im of the standard form we propose a more flexible class of conditional ims based on localization examples of local conditional ims in a bivariate normal model and a normal variance components model are also given

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

a note on modified estimative prediction limits and distributions in this paper a modification of the estimative prediction limit is derived for which the coverage probability equals the target value to third order accuracy under ordinary repeated sampling it provides an explicit approximate solution to the equation given by barndorff nielsen cox 1996 which determines an upper prediction limit for an unobserved random variable the corresponding predictive density is specified in a simple closed form and its main properties are emphasised

prior distributions from pseudo likelihoods in the presence of nuisance parameters consider a model parameterized by where is the parameter of interest the problem of eliminating the nuisance parameter can be tackled by resorting to a pseudo likelihood function l for namely a function of only and the data y with properties similar to those of a likelihood function if one treats l as a true likelihood the posterior distribution y l for can be considered where is a prior distribution on the goal of this article is to construct probability matching priors for a scalar parameter of interest only i e priors for which bayesian and frequentist inference agree to some order of approximation to be used in y when l is a marginal a conditional or a modification of the profile likelihood we show that is simply proportional to the square root of the inverse of the asymptotic variance of the pseudo maximum likelihood estimator the proposed priors are compared with the reference or jeffreys priors in four

exponential transformation models the class of exponential transformation models i e transformation models that are also exponential families is investigated it is shown by using standard exponential family theory that the group of transformations on the sample space generating the model induces affine transformation groups on the parameter space and on the range space of the minimal sufficient statistic the induced groups being in fact representations of the original transformation group the model function may be explicitly expressed in terms of these representations and it possesses a number of important properties the possibility of extending exponential transformation models to larger composite exponential transformation models is considered such extensions serving for instance the purpose of model control the inference for composite exponential transformation models is described in general terms and examples of such models are discussed the paper starts with an introduction to transformation models and a unified treatment of their distribution theory required in the subsequent discussion of the exponential transformation models

on the probable error of the ratio of two gamma means summary the two sample student s t interval can be derived by parametric bootstrap coverage calibration of the naive plug in interval that fails to account for the estimation of the common variance parameter the same technique results in an essentially exact confidence interval for the ratio of two means based on independent samples from gamma distributions the gamma problem is more difficult computationally because an exact pivot is not available however the computational burden of bootstrap calibration can be reduced to a routine level using the fact that the ratio of sample means is proportional to an exact f variate that is independent of the conditional maximum likelihood estimator of the shape parameter the density of the conditional estimator can be approximated with extreme accuracy using the p formula and a rejection sampler can be used to simulate from this approximate density this greatly simplifies bootstrap calibration in this context an alternative interval is obtained by inverting an exact test described by jensen 1986 simulation studies suggest that the two intervals have similar performance in terms of length and coverage in practice we recommend the bootstrap calibration method because it is much simpler to implement and is more versatile

efficiency of projected score methods in rectangular array asymptotics summary the paper considers a rectangular array asymptotic embedding for multistratum data sets in which both the number of strata and the number of within stratum replications increase and at the same rate it is shown that under this embedding the maximum likelihood estimator is consistent but not efficient owing to a non zero mean in its asymptotic normal distribution by using a projection operator on the score function an adjusted maximum likelihood estimator can be obtained that is asymptotically unbiased and has a variance that attains the cram6r rao lower bound the adjusted maximum likelihood estimator can be viewed as an approximation to the conditional maximum likelihood estimator

deviance residuals and normal scores plots summary we discuss the use of normal order statistics plots based on deviance residuals to check distributional assumptions in regression models continuous and discrete error distributions are considered as are censored data misspecified error distributions and discrimination between competing models are discussed with an example residual plots to detect inadequacies in normal linear regression models have a long history an example is the normal scores or rankit plot a plot of ordered residuals against normal order statistics which is used to detect outliers and to check distributional assumptions under or overdispersion in such a plot may also indicate a misspecified systematic component of the model the purpose of this paper is to discuss such plots for regressions with nonnormal errors such as generalized linear models for which deviance residuals are commonly used deviance residuals are known to be approximately normal in many cases in this paper we briefly describe the properties of rankit plots based on them for continuous distributions outline analogous results for discrete and censored data and describe a method for discriminating between competing models with different error distributions most roads lead to rome for the normal distribution in the sense that many definitions of residuals are functions of y jt o not so for other distributions for which various

approximate interval probabilities for models of parametric dimension 1 two parameterization invariant approximations to the conditional distribution function of the maximum likelihood estimator are considered the first is derived by partial integration of a formula for the conditional density given an ancillary statistic a technique which in addition yields a useful expression for the error term the second approximation is based on an adjusted version of the signed log likelihood ratio statistic a third approximation is a modification of the first that avoids the specification of an ancillary

a time series illustration of approximate conditional likelihood summary we consider the calculation of confidence intervals for the correlation parameter based on observations of a large number of short gaussian first order autoregressive processes with different means but common correlation and variance ordinary profile likelihood gives very misleading answers the approximate conditional likelihood of cox reid 1987 is calculated and shown to lead to consistent estimation of the correlation simulations verify that the usual asymptotic properties of the statistics derived from the conditional likelihood are quite satisfactory in this example the approximate conditional likelihood is the same as the marginal likelihood for the standardized residuals although this is not true in general

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12