saddlepoint approximations for estimating equations summary the saddlepoint method is used to approximate to the distribution of an estimator defined by an estimating equation two different approaches are available one of which is shown to be equivalent to the technique of field hampel 1982 two recent formulae for the tail probability due to lugannani rice 1980 and to robinson 1982 respectively which are uniformly accurate over the whole range of the estimator are compared numerically with the exact results and those computed by field hampel they are found to be of comparable accuracy while avoiding the use of numerical integration the most accurate is that of lugannani rice hampel 1974 introduced a new technique for approximating to the probability density of an estimator defined by an estimating equation it is an example of what he called small sample asymptotics where high accuracy is achieved for quite small sample sizes n even down to single figures in the original version it gave an approximation to the logarithmic derivative of the density function which was integrated numerically to get the density the distribution function was obtained by a second numerical integration which could then be used to renormalize both the density and the distribution function field hampel 1982 develop the technique in detail and compare its performance with that of other approximation methods as hampel had pointed out his approach is closely related to the saddlepoint method of daniels 1954 which was applied to sample means and ratios of means following the private communication referred to by field hampel 1982 p 31 it was realized that the first numerical integration was unnecessary and that hampel s approach could be shortened to give a direct approximation to the density which is in fact a saddlepoint approximation the purpose of the present paper is to extend the use of the saddlepoint method to estimating equations there are two distinct ways of doing this which lead to different approximations of similar accuracy one appears to be more convenient for approxi mating to tail probabilities by numerical integration and was used to compute the saddlepoint approximations quoted in field hampel s tables 1 and 2 the other gives the density directly in a form equivalent to their equation 4 3 which is then integrated

approximations of marginal tail probabilities and inference for scalar parameters summary in many situations inference for a scalar parameter in the presence of nuisance parameters requires integration of either a joint density of pivotal quantities or a joint posterior density for such inference accurate approximations of marginal tail probabilities are useful to avoid high dimensional integrals two tail probability approximations are developed in this paper numerical results given for conditional inference in location scale and linear regression models show the approximations to be generally accurate even for small sample sizes

modelling multiple time series via common factors we propose a new method for estimating common factors of multiple time series one distinctive feature of the new approach is that it is applicable to some nonstationary time series the unobservable nonstationary factors are identified by expanding the white noise space step by step thereby solving a high dimensional optimization problem by several low dimensional sub problems asymptotic properties of the estimation are investigated the proposed methodology is illustrated with both simulated and real datasets

parameter transformations for improved approximate confidence regions in nonlinear least squares in a previous paper it was shown that parameter effects nonlinearities of a nonlinear regression model experimental design parameterization combination can be quantified by means of a parameter effects curvature array a based on second derivatives of the model function in this paper the individual terms of a are interpreted and local compensation methods are suggested a method of computing the parameter effects array corresponding to a transformed set of parameters is given and we discuss how this result could be used to determine reparameterizations which have zero local parameter effects nonlinearity

a simple and accurate method for approximate conditional inference applied to exponential family models summary a simple method for approximate conditional inference is described the methodology is applied to natural exponential family models where it is shown to provide accurate approximations to fully conditional estimates the approximation technique can be applied much more generally than in this particular class of models the technique depends only on the construction of certain projected scores derived from higher order likelihood derivatives and their covariances and so can be used in many problems where it is relevant to control for the estimation of nuisance parameters

functional additive models in commonly used functional regression models the regression of a scalar or functional response on the functional predictor is assumed to be linear this means that the response is a linear function of the functional principal component scores of the predictor process we relax the linearity assumption and propose to replace it by an additive structure leading to a more widely applicable and much more flexible framework for functional regression models the proposed functional additive regression models are suitable for both scalar and functional responses the regularization needed for effective estimation of the regression parameter function is implemented through a projection on the eigenbasis of the covariance operator of the functional components in the model the use of functional principal components in an additive rather than linear way leads to substantial broadening of the scope of functional regression models and emerges as a natural approach because the uncorrelatedness of the functional pr

a pairwise interaction model for multivariate functional and longitudinal data functional data vectors consisting of samples of multivariate data where each component is a random function are encountered increasingly often but have not yet been comprehensively investigated we introduce a simple pairwise interaction model that leads to an interpretable and straightforward decomposition of multivariate functional data and of their variation into component specific processes and pairwise interaction processes the latter quantify the degree of pairwise interactions between the components of the functional data vectors while the component specific processes reflect the functional variation of a particular functional vector component that cannot be explained by the other components thus the proposed model provides an extension of the usual notion of a covariance or correlation matrix for multivariate vector data to functional data vectors and generates an interpretable functional interaction map the decomposition provided by the model can also serve as a basis for subsequent analysis such as study of the network structure of functional data vectors the decomposition of the total variance into componentwise and interaction contributions can be quantified by an formula see text like decomposition we provide consistency results for the proposed methods and illustrate the model by applying it to sparsely sampled longitudinal data from the baltimore longitudinal study of aging examining the relationships between body mass index and blood fats

exponential linear models a two pass procedure for saddlepoint approximation for an exponential linear model the saddlepoint method gives accurate approximations for the density of the minimal sufficient statistic or maximum likelihood estimate and for the corresponding distribution functions in this paper we describe a simple numerical procedure that constructs such approximations for a real parameter in an exponential linear model using only a two pass calculation on the observed likelihood function for the original data simple examples of the numerical procedure are discussed but we take the general accuracy of the saddlepoint procedure as given an immediate application of this is to exponential family models where inference for a component of the canonical parameter is to be based on the conditional density of the corresponding component of the sufficient statistic given the remaining components this conditional density is also of exponential family form but its functional form and cumulant generating function may not be accessible the procedure is applied to the corresponding likelihood approximated as the full likelihood divided by an approximate marginal likelihood obtained from barndorff nielsen s formula a double saddlepoint approximation provides another means of bypassing this difficulty the computational procedure is also examined as a numerical procedure for obtaining the saddlepoint approximation to the fourier inversion of a characteristic function as such it is a two pass calculation on a table of the cumulant generating function

confidence regions for parameter subsets in nonlinear regression on utilise la methode a deux niveaux proposee par hamilton watts et bates 1982 pour obtenir des approximations ameliorees des regions de confiance

functional data analysis for sparse longitudinal data we propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data the method aims at irregularly spaced longitudinal data where the number of repeated measurements available per subject is small in contrast classical functional data analysis requires a large number of regularly spaced measurements per subject we assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random subject specific trajectory plus measurement errors basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories and the estimation of the variance of the measurement errors the eigenfunction basis is estimated from the data and functional principal components score estimates are obtained by a conditioning step this conditional estimation method is conceptually simple and straightforward to implement

index models for sparsely sampled functional data the regression problem involving functional predictors has many important applications and a number of functional regression methods have been developed however a common complication in functional data analysis is one of sparsely observed curves that is predictors that are observed with error on a small subset of the possible time points such sparsely observed data induce an errors in variables model where one must account for measurement error in the functional predictors faced with sparsely observed data most current functional regression methods simply estimate the unobserved predictors and treat them as fully observed thus failing to account for the extra uncertainty from the measurement error we propose a new functional errors in variables approach sparse index model functional estimation simfe which uses a functional index model formulation to deal with sparsely observed predictors simfe has several advantages over more traditional methods first the index model implements a nonline

maximal space regularity for abstract linear non autonomous parabolic equations the linear non autonomous evolution equation u t a t u t f t t e 0 t with the initial datum u o x in the space c o t e where e is a banach space and a t is a family of infinitesimal generators of bounded analytic semigroups is considered the domains d t are supposed constant in t and possibly not dense in e maximal regularity of the strict and classical solutions i e regularity of u and a u with values in the interpolation spaces d o l co and da b between d o and e is studied a characterization of such spaces in a concrete case is also given 0 1985 academic press inc

robust statistical modeling using the t distribution abstract the t distribution provides a useful extension of the normal for statistical modeling of data sets involving errors with longer than normal tails an analytical strategy based on maximum likelihood for a general model with multivariate t errors is suggested and applied to a variety of problems including linear and nonlinear regression robust estimation of the mean and covariance matrix with missing data unbalanced multivariate repeated measures data multivariate modeling of pedigree data and multivariate nonlinear regression the degrees of freedom parameter of the t distribution provides a convenient dimension for achieving robust statistical inference with moderate increases in computational complexity for many models estimation of precision from asymptotic theory and the bootstrap is discussed and graphical methods for checking the appropriateness of the t distribution are presented

mean loglikelihood and higher order approximations summary higher order approximations to p values can be obtained from the log likelihood function and a reparameterization that can be viewed as a canonical parameter in an exponential family approximation to the model this approach clarifies the connection between skovgaard 1996 and fraser et al 1999a and shows that the skovgaard approximation can be obtained directly using the mean log likelihood function

functional linear regression analysis for longitudinal data we propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data where both the predictor and response are functions of a covariate such as time predictor and response processes have smooth random trajectories and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects in longitudinal studies the number of repeated measurements per subject is often small and may be modeled as a discrete random number and accordingly only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit we propose a functional regression approach for this situation using functional principal component analysis where we estimate the functional principal component scores through conditional expectations this allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory the resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories asymptotic properties for a sample of n subjects are investigated under mild conditions as n oo and we obtain consistent estimation for the regression function besides convergence results for the components of functional linear regression such as the regression parameter function we construct asymptotic pointwise confidence bands for the predicted trajectories a functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced extending the standard r 2 to the functional case the proposed methods are illustrated with a simulation study longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index

dynamical correlation for multivariate longitudinal data nonparametric methodology for longitudinal data analysis is becoming increasingly popular the analysis of multivariate longitudinal data where data on several time courses are recorded for each subject has received considerably less attention despite its importance for practical data analysis in particular there is a need for measures and estimates to capture dependency between the components of vector valued longitudinal data we propose and analyze a simple and effective nonparametric method to quantify the covariation of components of multivariate longitudinal observations which are viewed as realizations of a random process this includes the notion of a correlation between derivatives and time shifted versions the concept of dynamical correlation is based on a scalar product obtained from pairs of standardized smoothed curves the proposed method can be used when observation times are irregular and not matching between subjects or between responses within a subject for higher dimensional dat

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

accounting for intrinsic nonlinearity in nonlinear regression parameter inference regions joint confidence and likelihood regions for the parameters in nonlinear regression models can be defined using the geometric concepts of sample space and solution locus using a quadratic approximation to the solution locus instead of the usual linear approximation it is shown that these inference regions correspond to ellipsoids on the tangent plane at the least squares point accurate approximate inference regions can be obtained by mapping these ellipsoids into the parameter space and measures of the effect of intrinsic nonlinearity on inference can be based on the difference between the tangent plane ellipsoids and the sphere which would be obtained using a linear approximation

approximate conditional inference in generalized linear models on developpe des approximations precises et facilement calculables pour les densites conditionnelles et les distributions des statistiques exhaustives dans les modeles lineaires generalises avec des fonctions de liaison canoniques

on the abstract cauchy problem of parabolic type in spaces of continuous functions on etudie l existence l unicite et les proprietes de regularite des solutions du probleme de cauchy u t au r f t o t t u o u 0 ou a d a ce e est un operateur lineaire clos dans l espace de banach e

bounded solutions of linear periodic abstract parabolic equations on etudie le comportement asymptotique de l operateur d evolution parabolique g t s associe a une famille de generateurs de sous groupes analytiques a t t r dans un espace de banach general sous l hypothese de periodicite a t a t t on trouve de nouveaux resultats de regularite maximale pour les solutions bornees d equations paraboliques non homogenes abstraites dans des intervalles de temps non bornes

linear manifold modelling of multivariate functional data summary multivariate functional data are increasingly encountered in data analysis whereas statistical models for such data are not well developed yet motivated by a case study where one aims to quantify the relationship between various longitudinally recorded behaviour intensities for drosophila flies we propose a functional linear manifold model this model reflects the functional dependence between the components of multivariate random processes and is defined through data determined linear combinations of the multivariate component trajectories which are characterized by a set of varying coefficient functions the time varying linear relationships that govern the components of multivariate random functions yield insights about the underlying processes and also lead to noise reduced representations of the multivariate component trajectories the functional linear manifold model proposed is put to the task for an analysis of longitudinally observed behavioural patterns of flying feeding walking and resting over the lifespan of drosophila flies and is also investigated in simulations

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

canonical correlation analysis when the data are curves summary it is not immediately straightforward to extend canonical correlation analysis to the context of functional data analysis where the data are themselves curves or functions the obvious approach breaks down and it is necessary to use a method involving smoothing in some way such a method is introduced and discussed with reference to a data set on human gait the breakdown of the unsmoothed method is illustrated in a practical context and is demonstrated theoretically a consistency theorem for the smoothed method is proved in an increasing number of problems in a wide range of fields the data observed are not the univariate or multivariate observations of classical statistics but are func tions observed continuously ramsay and dalzell 1991 give the name functional data analysis to the analysis of data of this kind in most cases the observations will be functions of time or a closely related variable but there are clearly applications where the functions are surfaces observed over two or three dimensional space the motivating example for the present paper is a set of data on children s gait collected by the motion analysis laboratory at children s hospital san diego california see olshen et al 1989 for full details for each of a number of children several angles made by the child s joints knee hip etc are observed during the child s gait cycle one aim of the study is to gain understanding of the gait cycle of a normal child to make comparisons with children suffering from walking diffi culties these data motivated rice and silverman 1991 to discuss the extension of principal component analysis to the functional setting and to explain how smoothing can be incorporated into the analysis in a natural way in a substantial complemen tary paper ramsay and dalzell 1991 discuss different approaches to regression and principal component analysis in functional data analysis illustrating their work by a meteorological example the insights of functional data analysis may be helpful in chemometrics for instance a standard chemometric problem is the analysis of spectra observed in chromatography and these are to all intents and purposes functional observations another obvious area of potential relevance is the analysis of growth curve data the focus in the present paper will be on canonical correlation analysis cca in the gait example one might ask how variability in the knee angle cycle is related to

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

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

approximations of marginal tail probabilities for a class of smooth functions with applications to bayesian and conditional inference summary this paper presents an asymptotic approximation of marginal tail probabilities for a real valued function of a random vector where the function has continuous gradient that does not vanish at the mode of the joint density of the random vector this approximation has error 0 n 312 and improves upon a related standard normal approximation which has error 0 n 1 derivation involves the application of a tail probability formula given by diciccio field fraser 1990 to an approximation of a marginal density derived by tierney kass kadane 1989 the approximation can be applied for bayesian and conditional inference as well as for approximating sampling distributions and the accuracy of the approximation is illustrated through several numerical examples related to such applications in the context of conditional inference we develop refinements of the standard normal approximation to the distribution of two different signed root likelihood ratio statistics for a component of the natural parameter in exponential families

cα regularity for non autonomous linear integrodifferential equations of parabolic type abstract linear integrodifferential equations in general banach space are studied and applications are given to linear integrodifferential partial differential equations

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

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

small sample asymptotic distributions of m estimators of location summary asymptotic formulae for the distribution of m estimators i e maximum likelihood type estimators of location including the arithmetic mean are derived which numerical studies show to give relative errors for densities and tail areas of the order of magnitude of 1 down to sample sizes 3 and 4 even in the extreme tails the paper is the continuation of earlier work by the second author and is also closely related to daniels s work on the saddlepoint approximation the method consists in expanding the derivative of the logarithm of the unstandardized density of the estimator in powers of 1 n at each point using recentring by means of conjugate distributions this method yields a unified point of view for the comparison of other asymptotic methods namely saddlepoint method edgeworth expansion and large deviations approach which are also compared numerically

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

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

nonnormal linear regression an example of significance levels in high dimensions summary analysis of nonnormal linear models leads to an initial conditioning on the standardized residuals giving an unnormed density on rk where k is the number of parameters to obtain an observed level of significance for a single parameter it is then necessary to calculate a marginal probability thus requiring integration in k dimensions in this paper a conditional approach to evaluating the observed level of significance is developed and an importance sampling technique is used to improve the approximation and assess the accuracy of the conditional approximation to the marginal observed level a further approximation based on the invariant version fraser 1990 of the lugannani rice 1980 formula is also proposed the approach extends to the evaluation of real pivots and to bayesian inference for a single parameter component