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

high order asymptotic expansions for robust tests summary this paper provides high order asymptotic expansions for the m ratio and the signedroot m ratio robust test statistics which allow one to compute accurate approximations to significance or critical values using the edgeworth approximation the bartlett correction the variance correction or the saddlepoint approximation specific results are obtained for the linear regression model with the huber m estimator a monte carlo study illustrates the numerical accuracy of these approximations with respect to the usual first order approximations

saddlepoint approximations for the bartlett nanda pillai trace statistic in multivariate analysis summary the null distribution of the bartlett nanda pillai trace statistic is shown to be equivalent to the conditional distribution of canonical sufficient statistics in a multiparameter exponential family setting as such sequential and double saddlepoint approximations are used to approximate the distribution the existence of tabulated percentiles for the distribution allows for comparison of the accuracy of the various saddlepoint approximations the ability of the approximations to accommodate large numbers of nuisance parameters is also revealed with numerical studies the bartlett nanda pillai trace statistic is one of the four principal test statistics used in the multivariate analysis of variance p values of tests based on it are computed from its null distribution we show in 2 that this null distribution is equivalent to a conditional distribution involving components of the canonical sufficient statistic in the context of a certain multiparameter regular exponential family such equivalence allows for the possibility of saddlepoint approximation for both the null density and distribution function so that highly accurate approximations for p values can be easily obtained section 3 reviews the various saddlepoint methods that we shall base approximations upon sequential saddlepoint methods introduced by fraser reid wong 1991 are used to compute a sequential density approximation as well as distributional approximations based on the lugannani rice 1980 formula and the r approximation of bamdorff nielsen 1986 1991 double saddlepoint analogues to the above methods are also computed a double saddlepoint density approximation bamdorff nielsen cox 1979 takes the explicit form in 3 8 below as a modification to a rescaled beta density this explicit form is due to the explicit solution 4 11 that arises from the double saddlepoint equation double saddlepoint distributional approximations based on skovgaard 1987 and the r approximation bamdorff nielsen 1986 1991 are also computed the numerical comparisons of 6 reveal that the double saddlepoint distributional approximations of skovgaard 1987 and bamdorff nielsen 1986 1991 are grossly

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

comparing the means and variances of a bivariate log normal distribution for a bivariate log normal distribution a confidence interval is developed for the ratio of the means the generalized confidence interval approach is used for this purpose and the procedure is applicable regardless of the sample size it is also noted that the same approach can be used to obtain a confidence interval for the ratio of the variances a modified signed log likelihood ratio procedure is also described for computing confidence intervals the coverage probabilities of the proposed confidence intervals are estimated by monte carlo and the generalized confidence intervals are found to exhibit satisfactory performance even for small sample sizes numerical results also show that the corresponding test procedures provide larger power compared with the modified signed log likelihood ratio test two examples are given one dealing with the comparison of the means and variances of health care costs and the other dealing with testing mean equivalence in quantitative assays copyright 2007 john wiley sons ltd

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

a comparison of the efron hinkley ancillary and the likelihood ratio ancillary in a particular example two approximate ancillaries the efron hinkley ancillary and the signed likelihood ratio are compared in a specific example by means of their level curves their marginal densities and the conditional and unconditional likelihood functions it is shown that in the present model the latter of the two ancillaries is the better it is almost exactly distribution constant and has more stable conditional likelihood functions

higher order asymptotics and the likelihood principle one parameter models summary inferences based on higher order asymptotics in contrast to those based on standard first order methods approximate exact inferences closely enough to depend on the reference set we use this feature to quantify the effect of the reference set on exact inferences in particular the difference in p values for two reference sets can be at most o n and it is seen that one can often identify the coefficient of n although this is useful from a foundational viewpoint the question arises of whether higher order adjustments to firstorder p values reflect only the choice of reference set it is indicated that for one parameter problems this may largely be the case but that when there are nuisance parameters the situation is likely to be quite different the nuisance parameter setting is not investigated thoroughly in this paper but we believe that in that case the most important part of the higher order corrections largely conforms to the likelihood principle

saddlepoint approximations to the cdf of some statistics with nonnormal limit distributions abstract in standard saddlepoint approximations to the cumulative distribution function of a random variable the normal distribution has appeared to play a special role in this article we consider what happens when the normal base distribution is replaced by an arbitrary base distribution generalized versions of several standard formulas are presented the choice of a chi squared base or an inverse gaussian base is then considered in detail the generalized approximations are compared in two examples a linear combination of chi squared variables and the first passage time distribution for a random walk the former example considers approximations using the chi squared base that are slightly more accurate than their normal based counterparts in the latter example approximations based on the inverse gaussian are considerably more accurate than their normal based counterparts

residuals in generalized linear models abstract generalized linear models are regression type models for data not normally distributed appropriately fitted by maximum likelihood rather than least squares typical examples are models for binomial or poisson data with a linear regression model for a given ordinarily nonlinear function of the expected values of the observations use of such models has become very common in recent years and there is a clear need to study the issue of appropriate residuals to be used for diagnostic purposes several definitions of residuals are possible for generalized linear models the statistical package glim baker and nelder 1978 routinely prints out residuals where v is the function relating the variance to the mean of y and is the maximum likelihood estimate of the ith mean as fitted to the regression model these residuals are the signed square roots of the contributions to the pearson goodness of fit statistic another choice of residual is the signed square root of the contribution to the devia

on secant approximations to cumulative distribution functions summary we investigate the properties of an approximation called the secant approximation to the cumulative distribution function where the density is of a broad parametric class formulae for higher order terms are derived that give the approximation explicitly in terms of functions that define the density in all cases examined the first order secant approximation is monotonic and has the correct limits at both ends of the support resulting in the important property that the approximation itself is a cumulative distribution function we give conditions under which the approximation has bounded relative errors throughout the support the superiority of the secant approximation is illustrated in the particular case of the skew t family of distributions also known as pearson s type iv family a collection of formulae useful for the systematic evaluation of these approximations is given in appendix 2

confidence intervals of effect size in randomized comparative parallel group studies we have presented a new likelihood based approach for constructing confidence intervals of effect size that are applicable to small samples we also conduct a simulation study to compare the coverage probability of the new likelihood based method with other three methods proposed by hedges and olkin and kraemer and paik simulation studies show that the confidence interval generated by the modified signed log likelihood ratio method possesses essentially exact coverage probabilities even for small samples although the coverage probabilities are consistently but slightly less than the nominal level the methods are also applied to two examples

screening based on the risk of cancer calculation from bayesian hierarchical changepoint and mixture models of longitudinal markers the standard approach to early detection of disease with a quantitative marker is to set a population based fixed reference level for making further individual screening or referral decisions for many types of disease including prostate and ovarian cancer additional information is contained in the subject specific temporal behavior of the marker which exhibits a characteristic alteration early in the course of the disease in this article we derive a bayesian approach to screening based on calculation of the posterior probability of disease given longitudinal marker levels the method is motivated by a randomized ovarian cancer screening trial in the united kingdom comprising 22 000 women screened over 4 years with an additional 5 years of follow up on average levels of the antigen ca125 were recorded annually in the screened arm ca125 profiles of cases and controls from the u k trial are modeled using hierarchical changepoint and mixture models posterior distributions are calculated using markov

inferences on the mean response in a log regression model the generalized variable approach this paper proposes a novel approach for the confidence interval estimation and hypothesis testing of the mean response in a log regression model using the concept of generalized variable simulation studies demonstrate that the proposed confidence intervals have satisfying coverage probabilities the proposed approach can be simply carried out by a few simulation steps therefore it is an ideal candidate for making inferences about the mean response in a log regression model

improving on exact tests by approximate conditioning summary the often conservative nature for discrete data of so called exact tests seems usually the result of unnecessarily precise conditioning we consider avoiding this by conditioning only approximately on the sufficient statistics for nuisance parameters modest relaxation of conditioning results in small loss in terms of the rationale for conditional inference but can greatly reduce the difficulties caused by discreteness exact calculation of p values based on approximate conditioning is possible but unattractive both in terms of the amount of calculation involved and in requiring explicit specification of the extent to which conditioning is to be relaxed it is shown that there is a highly accurate easily computed and very natural asymptotic approximation that avoids these difficulties

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

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

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

optimum thresholding using mean and conditional mean squared error abstract we consider a univariate semimartingale model for the logarithm of an asset price containing jumps having possibly infinite activity the nonparametric threshold estimator i v n of the integrated variance i v 0 t s 2 d s proposed in mancini 2009 is constructed using observations on a discrete time grid and precisely it sums up the squared increments of the process when they are below a threshold which depends on the observation time step and sometimes model parameters or latent variables that need to be estimated all the threshold functions satisfying given conditions allow asymptotically consistent estimates of i v however the finite sample properties of i v n can depend on the specific choice of the threshold we aim here at optimally selecting the threshold by minimizing either the estimation mean squared error mse or the conditional mean squared error cmse the last criterion allows to reach a threshold which is optimal not in mean but for the specific volatility and jumps paths at hand a parsimonious characterization of the optimum is established which turns out to be asymptotically proportional to the levy s modulus of continuity of the underlying brownian motion moreover minimizing the cmse enables us to propose a novel implementation scheme for approximating the optimal threshold monte carlo simulations illustrate the superior performance of the proposed method

likelihood based confidence intervals for a log normal mean to construct a confidence interval for the mean of a log normal distribution in small samples we propose likelihood based approaches the signed log likelihood ratio and modified signed log likelihood ratio methods extensive monte carlo simulation results show the advantages of the modified signed log likelihood ratio method over the signed log likelihood ratio method and other methods in particular the modified signed log likelihood ratio method produces a confidence interval with a nearly exact coverage probability and highly accurate and symmetric error probabilities even for extremely small sample sizes we then apply the methods to two sets of real life data

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

small sample inference for the comparison of means of log normal distributions we propose a likelihood based test for comparing the means of two or more log normal distributions with possibly unequal variances a modification to the likelihood ratio test is needed when sample sizes are small the performance of the proposed procedures is compared with the f ratio test using monte carlo simulations

highly accurate likelihood analysis for the seemingly unrelated regression problem the linear and nonlinear seemingly unrelated regression problem with general error distribution is analyzed using recent likelihood theory that arguably provides the definitive distribution for assessing a scalar parameter this involves implicit but well defined conditioning and marginalization for determining intrinsic measures of departure highly accurate p values are obtained for the key difference between two regression coefficients of central interest the p value gives the statistical position of the data with respect to the key parameter the theory and the results indicate that this methodology provides substantial improvement on first order likelihood procedures both in distributional accuracy and in precise measurement of the key parameter

the modified signed likelihood statistic and saddlepoint approximations summary for a number of tests in exponential families we show that the use of a normal approximation to the modified signed likelihood ratio statistic r is equivalent to the use of a saddlepoint approximation this is also true in a large deviation region where the signed likelihood ratio statistic r is of order in

practical small sample asymptotics for distributions used in life data analysis fraser and lawless discussed exact conditional intervals for the parameters and quantiles of the location scale model when complete data are used moreover lawless extended the exact method to failure censored type ii censored data nevertheless the exact intervals are difficult to obtain in practice and are unavailable under time censoring type i censoring as a consequence approximate large sample intervals are widely used in this article a likelihood based third order procedure is developed the method does not require explicit nuisance parameterization and can be easily implemented into algebraic computational packages numerical examples are presented to show the accuracy of the method even when the sample size is small

an accurate method for approximate conditional and bayesian inference about linear regression models from censored data summary a method to approximate marginal posterior distribution functions of regression coefficients and scale parameters is presented it is based on a general asymptotic formula of diciccio field fraser 1990 for approximating marginal tail probabilities in developing the method it is shown that the accuracy of the approximation of diciccio et al can be substantially increased in certain cases by using a suitable change of variables and a general approach to deriving such transformations is considered

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

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

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

interval estimates for the ratio and difference of two lognormal means health research often gives rise to data that follow lognormal distributions in two sample situations researchers are likely to be interested in estimating the difference or ratio of the population means several methods have been proposed for providing confidence intervals for these parameters however it is not clear which techniques are most appropriate or how their performance might vary additionally methods for the difference of means have not been adequately explored we discuss in the present article five methods of analysis these include two methods based on the log likelihood ratio statistic and a generalized pivotal approach additionally we provide and discuss the results of a series of computer simulations finally the techniques are applied to a real example

on the common mean of several inverse gaussian distributions based on a higher order likelihood method an interval estimation method for the common mean of several heterogeneous inverse gaussian ig populations is discussed the proposed method is based on a higher order likelihood based procedure the merits of the proposed method are numerically compared with the signed log likelihood ratio statistic two generalized pivot quantities and the simple t test method with respect to their expected lengths coverage probabilities and type i errors numerical studies show that the coverage probabilities of the proposed method are very accurate and type i errors are close to the nominal level 05 even for very small samples the methods are also illustrated with two examples