bayesian modeling of the covariance structure for irregular longitudinal data using the partial autocorrelation function in long term follow up studies irregular longitudinal data are observed when individuals are assessed repeatedly over time but at uncommon and irregularly spaced time points modeling the covariance structure for this type of data is challenging as it requires specification of a covariance function that is positive definite moreover in certain settings careful modeling of the covariance structure for irregular longitudinal data can be crucial in order to ensure no bias arises in the mean structure two common settings where this occurs are studies with outcome dependent follow up and studies with ignorable missing data outcome dependent follow up occurs when individuals with a history of poor health outcomes had more follow up measurements and the intervals between the repeated measurements were shorter when the follow up time process only depends on previous outcomes likelihood based methods can still provide consistent estimates of the regression parameters given that both the mean and covariance structures of the irregular longitudinal data are correctly specified and no model for the follow up time process is required for ignorable missing data the missing data mechanism does not need to be specified but valid likelihood based inference requires correct specification of the covariance structure in both cases flexible modeling approaches for the covariance structure are essential in this paper we develop a flexible approach to modeling the covariance structure for irregular continuous longitudinal data using the partial autocorrelation function and the variance function in particular we propose semiparametric non stationary partial autocorrelation function models which do not suffer from complex positive definiteness restrictions like the autocorrelation function we describe a bayesian approach discuss computational issues and apply the proposed methods to cd4 count data from a pediatric aids clinical trial 2015 the authors statistics in medicine published by john wiley sons ltd

multilinear schur multipliers and schatten properties of operator taylor remainders abstract we establish sharp conditions on scalar functions and perturbations that guarantee schatten summability of n th order operator taylor remainders in the special case of dimension one our estimates of these remainders deliver well known classical estimates of scalar taylor remainders we prove that if a scalar function f is in the set c n and a perturbation is in the p th schatten class s p p n then the respective n th order operator taylor remainder is an element of s p n and has an estimate like the one in 16 we construct examples of f c n and perturbations in s n such that the n th order taylor remainder of the respective operator function is not in s 1 our construction relies in particular on novel dimension dependent estimates for schatten norms of multilinear schur multipliers from below that are of interest in their own right our results apply to both self adjoint and unitary operators

trace formulas for relative schatten class perturbations abstract we derive trace formulas for a pair of self adjoint operators h v and h under the assumption that h i 1 v is in a schatten class this extends the trace formulas of 8 where v alone is assumed to be in a schatten class our trace formulas apply in particular in the setting of differential operators and are based on taylor like approximations of operator functions this significantly improves non taylor based trace formulas of 10

joint mean covariance models with applications to longitudinal data unconstrained parameterisation summary we provide unconstrained parameterisation for and model a covariance using covariates the cholesky decomposition of the inverse of a covariance matrix is used to associate a unique unit lower triangular and a unique diagonal matrix with each covariance matrix the entries of the lower triangular and the log of the diagonal matrix are unconstrained and have meaning as regression coefficients and prediction variances when regressing a measurement on its predecessors an extended generalised linear model is introduced for joint modelling of the vectors of predictors for the mean and covariance subsuming the joint modelling strategy for mean and variance heterogeneity gabriel s antedependence models dempster s covariance selection models and the class of graphical models the likelihood function and maximum likelihood estimators of the covariance and the mean parameters are studied when the observations are normally distributed applications to modelling nonstationary dependence structures and multivariate data are discussed and illustrated using real data a graphical method similar to that based on the correlogram in time series is developed and used to identify parametric models for nonstationary covariances

asymptotic expansions for trace functionals abstract we obtain taylor approximations for functionals v tr f h 0 v defined on the bounded self adjoint operators where h 0 is a self adjoint operator with compact resolvent and f is a sufficiently nice scalar function relaxing assumptions on the operators made in 17 and derive estimates and representations for the remainders of these approximations

modelling of covariance structures in generalised estimating equations for longitudinal data summary when used for modelling longitudinal data generalised estimating equations specify a working structure for the within subject covariance matrices aiming to produce efficient parameter estimators however misspecification of the working covariance structure may lead to a large loss of efficiency of the estimators of the mean parameters in this paper we propose an approach for joint modelling of the mean and covariance structures of longitudinal data within the framework of generalised estimating equations the resulting estimators for the mean and covariance parameters are shown to be consistent and asymptotically normally distributed real data analysis and simulation studies show that the proposed approach yields efficient estimators for both the mean and covariance parameters

trace formulas for resolvent comparable operators abstract we derive trace formulas for self adjoint perturbations of self adjoint operators such that the difference of the resolvents of the initial and perturbed operators is an element of a schatten von neumann ideal s n n n this substantially extends the result of 11 for perturbations corresponding to the resolvent difference in the hilbert schmidt class s 2 and the result of 13 for perturbations in s n n n

multilevel time series models with applications to repeated measures data the analysis of repeated measures data can be conducted efficiently using a two level random coefficients model a standard assumption is that the within individual level 1 residuals are uncorrelated in some cases especially where measurements are made close together in time this may not be reasonable and this additional correlation structure should also be modelled a time series model for such data is proposed which consists of a standard multilevel model for repeated measures data augmented by an autocorrelation model for the level 1 residuals first and second order autoregressive models are considered in detail together with a seasonal component both discrete and continuous time are considered and it is shown how the autocorrelation parameters can themselves be structured in terms of further explanatory variables the models are fitted to a data set consisting of repeated height measurements on children

multi parameter regression survival modeling an alternative to proportional hazards summary it is standard practice for covariates to enter a parametric model through a single distributional parameter of interest for example the scale parameter in many standard survival models indeed the well known proportional hazards model is of this kind in this article we discuss a more general approach whereby covariates enter the model through more than one distributional parameter simultaneously e g scale and shape parameters we refer to this practice as multi parameter regression mpr modeling and explore its use in a survival analysis context we find that multi parameter regression leads to more flexible models which can offer greater insight into the underlying data generating process to illustrate the concept we consider the two parameter weibull model which leads to time dependent hazard ratios thus relaxing the typical proportional hazards assumption and motivating a new test of proportionality a novel variable selection strategy is introduced for such multi parameter regression models it accounts for the correlation arising between the estimated regression coefficients in two or more linear predictors a feature which has not been considered by other authors in similar settings the methods discussed have been implemented in the mpr package in r

continuous versus discrete frequency changes different detection mechanisms sek and moore j acoust soc am 106 351 359 1999 and lyzenga et al j acoust soc am 116 491 501 2004 found that the just noticeable frequency difference between two pure tones relatively close in time is smaller when these tones are smoothly connected by a frequency glide than when they are separated by a silent interval this glide effect was interpreted as evidence that frequency glides can be detected by a specific auditory mechanism not involved in the detection of discrete time delayed frequency changes lyzenga et al argued in addition that the glide detection mechanism provides little information on the direction of frequency changes near their detection threshold the first experiment reported here confirms the existence of the glide effect but also shows that it disappears when the glide is not connected smoothly to the neighboring steady tones a second experiment demonstrates that the direction of a 750ms frequency glide can be perceptually identified as soon as the glide

functions of noncommuting self adjoint operators under perturbation and estimates of triple operator integrals abstract we define functions of noncommuting self adjoint operators with the help of double operator integrals we are studying the problem to find conditions on a function f on r 2 for which the map a b f a b is lipschitz in the operator norm and in schatten von neumann norms s p it turns out that for functions f in the besov class b 1 1 r 2 the above map is lipschitz in the s p norm for p 1 2 however it is not lipschitz in the operator norm nor in the s p norm for p 2 the main tool is triple operator integrals to obtain the results we introduce new haagerup like tensor products of l spaces and obtain schatten von neumann norm estimates of triple operator integrals we also obtain similar results for functions of noncommuting unitary operators

cholesky decompositions and estimation of a covariance matrix orthogonality of variance correlation parameters summary chen dunson 2003 have proposed a modified cholesky decomposition of the form e dll d for a covariance matrix where d is a diagonal matrix with entries proportional to the square roots of the diagonal entries of e and l is a unit lower triangular matrix solely determining its correlation matrix this total separation of variance and correlation is definitely a major advantage over the more traditional modified cholesky decomposition of the form ld2l pourahmadi 1999 we show that though the variance and correlation parameters of the former decomposition are separate they are not asymptotically orthogonal and that the estimation of the new parameters could be more demanding computationally we also provide statistical interpretation for the entries of l and d as certain moving average parameters and innovation variances and indicate how the existing likelihood procedures can be employed to estimate the new parameters

functions of almost commuting operators and an extension of the helton howe trace formula abstract let a and b be almost commuting i e the commutator ab ba belongs to trace class self adjoint operators we construct a functional calculus a b for functions in the besov class b 1 1 r 2 this functional calculus is linear the operators a b and a b almost commute for b 1 1 r 2 and a b u a v b whenever s t u s v t we extend the helton howe trace formula for arbitrary functions in b 1 1 r 2 the main tool is triple operator integrals with integrands in haagerup like tensor products of l spaces

detection of linear frequency glides as a function of frequency and duration this article compares frequency discrimination thresholds for four conditions using a two interval forced choice task 1 discrimination of a difference in center frequency for two pulsed tones dlf 2 discrimination of a steady tone from an up frequency glide with the same mean frequency up dl 3 discrimination of a steady tone from a down frequency glide with the same mean frequency down dl 4 discrimination of an up frequency glide from a down frequency glide with the same mean frequency upvsdown dl all glides were linear in hz s experiment 1 measured thresholds as a function of center frequency over the range 500 8000 hz in 1 oct steps for a fixed duration of 500 ms the variation of glide thresholds with frequency was similar to that for the dlfs experiment 2 measured thresholds as a function of duration over the range 50 750 ms for a fixed center frequency of 2000 hz the dlfs decreased monotonically with increasing duration the glide thresholds decreased as duration increased f

detection thresholds for sinusoidal frequency modulation an adaptive forced choice procedure was used to measure in four normal hearing subjects detection thresholds for sinusoidal frequency modulation as a function of carrier frequency fc from 250 to 4000 hz and modulation frequency fmod from 1 to 64 hz the results show that for a wide range of fmod values fc and fmod have almost independent effects on the thresholds when the thresholds are expressed as just noticeable frequency swings and plotted on a log scale in two subjects the effect of fc on the thresholds was compared to the effect of standard frequency on the frequency just noticeable differences jnd s of successive and steady tones in agreement with previous data h fastl j acoust soc am 63 275 277 1978 it was found that the two effects are significantly different if the frequency jnd s are measured with long duration tones however it was also found that the two effects are similar if the frequency jnd s are measured with 25 ms tones these results support the idea that

a moving average cholesky factor model in covariance modelling for longitudinal data we propose new regression models for parameterizing covariance structures in longitudinal data analysis using a novel cholesky factor the entries in this decomposition have a moving average and log innovation interpretation and are modelled as linear functions of covariates we propose efficient maximum likelihood estimates for joint mean covariance analysis based on this decomposition and derive the asymptotic distributions of the coefficient estimates furthermore we study a local search algorithm computationally more efficient than traditional all subset selection based on bic for model selection and show its model selection consistency thus a conjecture of pan mackenzie 2003 is verified we demonstrate the finite sample performance of the method via analysis of data on cd4 trajectories and through simulations

multidimensional operator multipliers we introduce multidimensional schur multipliers and characterise them generalising well known results by grothendieck and peller we define a multidimensional version of the two dimensional operator multipliers studied recently by kissin and shulman the multidimensional operator multipliers are defined as elements of the minimal tensor product of several c algebras satisfying certain boundedness conditions in the case of commutative c algebras the multidimensional operator multipliers reduce to continuous multidimensional schur multipliers we show that the multipliers with respect to some given representations of the corresponding c algebras do not change if the representations are replaced by approximately equivalent ones we establish a non commutative and multidimensional version of the characterisations by grothendieck and peller which shows that universal operator multipliers can be obtained as certain weak limits of elements of the algebraic tensor product of the corresponding c algebras

dynamic semiparametric bayesian models for genetic mapping of complex trait with irregular longitudinal data many phenomena of fundamental importance to biology and biomedicine arise as a dynamic curve such as organ growth and hiv dynamics the genetic mapping of these traits is challenged by longitudinal variables measured at irregular and possibly subject specific time points in which case nonnegative definiteness of the estimated covariance matrix needs to be guaranteed we present a semiparametric approach for genetic mapping within the mixture model setting by jointly modeling mean and covariance structures for irregular longitudinal data penalized spline is used to model the mean functions of individual quantitative trait locus qtl genotypes as latent variables whereas an extended generalized linear model is used to approximate the covariance matrix the parameters for modeling the mean covariances are estimated by mcmc using the gibbs sampler and the metropolis hastings algorithm we derive the full conditional distributions for the mean and covariance parameters and compute bayes factors to test the hypothesis about the existence of significant qtls we used the model to screen the existence of specific qtls for age specific change of body mass index with a sparse longitudinal data set the new model provides powerful means for broadening the application of genetic mapping to reveal the genetic control of dynamic traits copyright 2012 john wiley sons ltd

model diagnostic tests for selecting informative correlation structure in correlated data in the generalized method of moments approach to longitudinal data analysis unbiased estimating functions can be constructed to incorporate both the marginal mean and the correlation structure of the data increasing the number of parameters in the correlation structure corresponds to increasing the number of estimating functions thus building a correlation model is equivalent to selecting estimating functions this paper proposes a chi squared test to choose informative unbiased estimating functions we show that this methodology is useful for identifying which source of correlation it is important to incorporate when there are multiple possible sources of correlation this method can also be applied to determine the optimal working correlation for the generalized estimating equation approach

estimates and trace formulas for unitary and resolvent comparable perturbations abstract we prove estimates for schatten norms of multiple operator integrals with symbols arising from fourier series and as a consequence derive higher order trace formulas for functions given by absolutely convergent fourier series of unitary operators and for rational functions bounded on the real line of resolvent comparable self adjoint operators these results substantially extend the main results of 12 14 13 where the estimates were proved for multiple operator integrals with symbols arising from polynomials the trace formulas in the unitary case for analytic functions on the unit disc and the trace formulas in the resolvent comparable self adjoint case for rational functions bounded and analytic in the lower half plane our results rely crucially on an algebraic approach to symbols of multiple operator integrals developed in this paper to replace the prior restrictive analytic approach of 12

higher order spectral shift abstract we construct higher order spectral shift functions extending the perturbation theory results of m g krein m g krein on a trace formula in perturbation theory mat sb 33 1953 597 626 in russian and l s koplienko l s koplienko trace formula for perturbations of nonnuclear type sibirsk mat zh 25 1984 62 71 in russian translation in trace formula for nontrace class perturbations siberian math j 25 1984 735 743 on representations for the remainders of the first and second order taylor type approximations of operator functions the higher order spectral shift functions represent the remainders of higher order taylor type approximations they can be expressed recursively via the lower order in particular krein s and koplienko s ones we also obtain higher order spectral averaging formulas generalizing the birman solomyak spectral averaging formula the results are obtained in the semi finite von neumann algebra setting with the perturbation taken in the hilbert schmidt class of the algebra

resolution of peller s problem concerning koplienko neidhardt trace formulae a formula for the norm of a bilinear schur multiplier acting from the cartesian product s 2 s 2 of two copies of the hilbert schmidt classes into the trace class s 1 is established in terms of linear schur multipliers acting on the space s 1 of all compact operators using this formula we resolve peller s problem on koplienko neidhardt trace formulae namely we prove that there exist a twice continuously differentiable function f with a bounded second derivative a self adjoint unbounded operator a and a self adjoint operator b 2 s 2 such that f a b f a d dt f a tb t 0 2 s 1

a semiparametric approach to simultaneous covariance estimation for bivariate sparse longitudinal data estimation of the covariance structure for irregular sparse longitudinal data has been studied by many authors in recent years but typically using fully parametric specifications in addition when data are collected from several groups over time it is known that assuming the same or completely different covariance matrices over groups can lead to loss of efficiency and or bias nonparametric approaches have been proposed for estimating the covariance matrix for regular univariate longitudinal data by sharing information across the groups under study for the irregular case with longitudinal measurements that are bivariate or multivariate modeling becomes more difficult in this article to model bivariate sparse longitudinal data from several groups we propose a flexible covariance structure via a novel matrix stick breaking process for the residual covariance structure and a dirichlet process mixture of normals for the random effects simulation studies are performed to investigate the effectiveness of the proposed approach over more traditional approaches we also analyze a subset of framingham heart study data to examine how the blood pressure trajectories and covariance structures differ for the patients from different bmi groups high medium and low at baseline

higher order extension of lowner s theory operator k tone functions the new notion of operator matrix k tone functions is introduced which is a higher order extension of operator matrix monotone and convex functions differential properties of matrix k tone functions are shown characterizations properties and examples of operator k tone functions are presented in particular integral representations of operator k tone functions are given generalizing familiar representations of operator monotone and convex functions

on modelling mean covariance structures in longitudinal studies we exploit a reparameterisation of the marginal covariance matrix arising in longitudinal studies pourahmadi 1999 2000 to model jointly the mean and covariance structures in terms of three polynomial functions of time by reanalysing kenward s 1987 cattle data we compare model selection procedures based on regressogram estimation with these based on a global search of the model space using a bic based model selection criterion to identify the optimum degree triple of the three polynomials we show that the use of a saturated mean model is not optimal and explain why regressogram based model estimation may be misleading we also suggest a new computational method for finding the global optimum based on a criterion involving three pairwise saturated profile likelihoods

operator holder zygmund functions abstract it is well known that a lipschitz function on the real line does not have to be operator lipschitz we show that the situation changes dramatically if we pass to holder classes namely we prove that if f belongs to the holder class r with 0 1 then f a f b const a b for arbitrary self adjoint operators a and b we prove a similar result for functions f in the zygmund class 1 r for arbitrary self adjoint operators a and k we have f a k 2 f a f a k const k we also obtain analogs of this result for all holder zygmund classes r 0 then we find a sharp estimate for f a f b for functions f of class def f f const for an arbitrary modulus of continuity in particular we study moduli of continuity for which f a f b const a b for self adjoint a and b and for an arbitrary function f in we obtain similar estimates for commutators f a q q f a and quasicommutators f a q q f b finally we estimate the norms of finite differences j 0 m 1 m j m j f a j k for f in the class m that is defined in terms of finite differences and a modulus continuity of order m we also obtain similar results for unitary operators and for contractions

a bayesian semiparametric model for bivariate sparse longitudinal data mixed effects models have recently become popular for analyzing sparse longitudinal data that arise naturally in biological agricultural and biomedical studies traditional approaches assume independent residuals over time and explain the longitudinal dependence by random effects however when bivariate or multivariate traits are measured longitudinally this fundamental assumption is likely to be violated because of intertrait dependence over time we provide a more general framework where the dependence of the observations from the same subject over time is not assumed to be explained completely by the random effects of the model we propose a novel mixed model based approach and estimate the error covariance structure nonparametrically under a generalized linear model framework we use penalized splines to model the general effect of time and we consider a dirichlet process mixture of normal prior for the random effects distribution we analyze blood pressure data from the framingham heart study where body mass index gender and time are treated as covariates we compare our method with traditional methods including parametric modeling of the random effects and independent residual errors over time we conduct extensive simulation studies to investigate the practical usefulness of the proposed method the current approach is very helpful in analyzing bivariate irregular longitudinal traits copyright 2013 john wiley sons ltd

two chain rules for divided differences and faa di bruno s formula in this paper we derive two formulas for divided differences of a function of a function both formulas lead to other divided difference formulas such as reciprocal and quotient rules the two formulas can also be used to derive faa di bruno s formula and other formulas for higher derivatives of composite functions we also derive a divided difference version of faa di bruno s determinant formula

spectral shift function of higher order this paper resolves affirmatively koplienko s sib mat zh 25 62 71 1984 conjecture on existence of higher order spectral shift measures moreover the paper establishes absolute continuity of these measures and thus existence of the higher order spectral shift functions a spectral shift function of order n is the function n n h v such that operatorname tr biggl f h v sum k 0 n 1 frac 1 k frac d k dt k bigl f h tv bigr bigg t 0 biggr int mathbb r f n t eta n t dt for every sufficiently smooth function f where h is a self adjoint operator defined in a separable hilbert space and v is a self adjoint operator in the n th schatten von neumann ideal sn existence and summability of 1 and 2 were established by krein mat sb 33 597 626 1953 and koplienko sib mat zh 25 62 71 1984 respectively whereas for n 2 the problem was unresolved we show that n h v exists integrable and vert eta n vert l 1 mathbb r leq c n vert v vert s n n for some constant cn depending only on n our results for n rely on estimates for multiple operator integrals obtained in this paper our method also applies to the general semi finite von neumann algebra setting of the perturbation theory

estimates of internal templates for the detection of sequential tonal patterns in this experiment listeners detected sequential tonal patterns embedded in multitone multiburst random maskers the maskers consisted of eight 30ms bursts of random frequency tones the signal when present occupied the central six bursts and was centered at 1000hz the six sequential signal tones formed several spectro temporal patterns an equal frequency pattern three ascending patterns with frequency ranges spanning 0 5 1 and 2 equivalent rectangular bandwidths erbs and a random pattern with frequencies drawn at random from the range of 925 1075hz the total number of tones in each burst m was varied to determine detection threshold the detectability of the signal pattern declined as the frequency range of the signal pattern increased and when the signal was random relative weights as a function of time and frequency interpreted as listeners internal templates depended systematically on the properties of the signal pattern tested the templates indicated that when sensitivity was po

functions of operators under perturbations of class sp this is a continuation of our paper 2 we prove that for functions f in the holder class r and 1 p the operator f a f b belongs to sp whenever a and b are self adjoint operators with a b sp we also obtain sharp estimates for the schatten von neumann norms f a f b sp in terms of a b sp and establish similar results for other operator ideals we also estimate schatten von neumann norms of higher order differences j 0m 1 m j mj f a jk we prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions then we find necessary conditions on f for f a f b to belong to sq under the assumption that a b sp we also obtain schatten von neumann estimates for quasicommutators f a r rf b and introduce a spectral shift function and find a trace formula for operators of the form f a k 2f a f a k

a sliding two alternative forced choice paradigm for pitch discrimination studies that measure frequency discrimination often use 2 3 or 4 tones per trial this paper shows an investigation of a two alternative forced choice 2afc task in which each tone of a series is judged relative to the previous tone sliding 2afc potential advantages are a greater yield number of responses per unit time and a more uniform history of stimulation for the study of context effects or to relate time varying performance to cortical activity the new task was evaluated relative to a classic 2 tone per trial 2afc task with similar stimulus parameters for each task conditions with different stimulus parameters were compared the main results were as follows 1 thresholds did not differ significantly between tasks when similar parameters were used 2 thresholds did differ between conditions for the new task showing a deleterious effect of inserting relatively large steps in the frequency sequence 3 thresholds also differed between conditions for the classic task showing an adv

maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix the positive definiteness constraint is the most awkward stumbling block in modelling the covariance matrix pourahmadi s 1999 unconstrained parameterisation models covariance using covariates in a similar manner to mean modelling in generalised linear models the new covariance parameters have statistical interpretation as the regression coefficients and logarithms of prediction error variances corresponding to regressing a response on its predecessors in this paper the maximum likelihood estimators of the parameters of a generalised linear model for the covariance matrix their consistency and their asymptotic normality are studied when the observations are normally distributed these results along with the likelihood ratio test and penalised likelihood criteria such as bic for model and variable selection are illustrated using a real dataset

multiple operator integrals and higher operator derivatives abstract in this paper we consider the problem of the existence of higher derivatives of the function t a tk where is a function on the real line a is a self adjoint operator and k is a bounded self adjoint operator we improve earlier results by sten kin in order to do this we give a new approach to multiple operator integrals this approach improves the earlier approach given by sten kin we also consider a similar problem for unitary operators

covariance structure regularization via frobenius norm discrepancy abstract in many practical problems the underlying structure of an estimated covariance matrix is usually blurred due to random noise particularly when the dimension of the matrix is high hence it is necessary to filter the random noise or regularize the available covariance matrix in certain senses so that the covariance structure becomes clear in this paper we propose a new method for regularizing the covariance structure of a given covariance matrix by choosing an optimal structure from an available class of covariance structures the regularization is made in terms of minimizing the discrepancy defined by frobenius norm between the given covariance matrix and the class of covariance structures a range of potential candidate structures including the order 1 moving average structure compound symmetry structure order 1 autoregressive structure order 1 autoregressive moving average structure are considered simulation studies show that the proposed new approach is reliable in regularization of covariance structures the proposed approach is also applied to real data analysis in signal processing showing the usefulness of the proposed approach in practice

higher order s2 differentiability and application to koplienko trace formula abstract let a be a selfadjoint operator in a separable hilbert space k a selfadjoint hilbert schmidt operator and f c n r we establish that t f a t k f a is n times continuously differentiable on r in the hilbert schmidt norm provided either a is bounded or the derivatives f i i 1 n are bounded this substantially extends the results of 3 on higher order differentiability of in the hilbert schmidt norm for f in a certain wiener class as an application of the second order s 2 differentiability we extend the koplienko trace formula from the besov class b 1 2 r 20 to functions f for which the divided difference f 2 admits a certain hilbert space factorization

semiparametric mean covariance regression analysis for longitudinal data efficient estimation of the regression coefficients in longitudinal data analysis requires a correct specification of the covariance structure existing approaches usually focus on modeling the mean with specification of certain covariance structures which may lead to inefficient or biased estimators of parameters in the mean if misspecification occurs in this article we propose a data driven approach based on semiparametric regression models for the mean and the covariance simultaneously motivated by the modified cholesky decomposition a regression spline based approach using generalized estimating equations is developed to estimate the parameters in the mean and the covariance the resulting estimators for the regression coefficients in both the mean and the covariance are shown to be consistent and asymptotically normally distributed in addition the nonparametric functions in these two structures are estimated at their optimal rate of convergence simulation studies and a real data analysis show

an extension of the koplienko neidhardt trace formulae abstract koplienko sibirsk mat zh 25 5 1984 62 found a trace formula for perturbations of self adjoint operators by operators of hilbert schmidt class s 2 a similar formula in the case of unitary operators was obtained by neidhardt math nachr 138 1988 7 in this paper we improve their results and obtain sharp conditions under which the koplienko neidhardt trace formulae hold

a joint modelling approach for longitudinal studies in longitudinal studies it is of fundamental importance to understand the dynamics in the mean function variance function and correlations of the repeated or clustered measurements for modelling the covariance structure cholesky type decomposition based approaches have been demonstrated to be effective however parsimonious approaches for directly revealing the correlation structure between longitudinal measurements remain less well explored and existing joint modelling approaches may encounter difficulty in interpreting the covariation structure we propose a novel joint mean variance correlation modelling approach for longitudinal studies by applying hyperspherical co ordinates we obtain an unconstrained parameterization for the correlation matrix that automatically guarantees its positive definiteness and we develop a regression approach to model the correlation matrix of the longitudinal measurements by exploiting the parameterization the modelling framework proposed is parsimonious interpretable and flexible for analysing longitudinal data extensive data examples and simulations support the effectiveness of the approach proposed

modelling covariance structure in bivariate marginal models for longitudinal data summary it can be more challenging to efficiently model the covariance matrices for multivariate longitudinal data than for the univariate case due to the correlations arising between multiple responses the positive definiteness constraint and the high dimensionality are further obstacles in covariance modelling in this paper we develop a data based method by which the parametersinthecovariancematricesarereplacedbyunconstrainedandinterpretableparameterswith reduced dimensions the maximum likelihood estimators for the mean and covariance parameters are shown to be consistent and asymptotically normally distributed simulations and real data analysis show that the new approach performs very well even when modelling bivariate nonstationary dependence structures

estimation of covariance matrix of multivariate longitudinal data using modified choleksky and hypersphere decompositions linear models are typically used to analyze multivariate longitudinal data with these models estimating the covariance matrix is not easy because the covariance matrix should account for complex correlated structures the correlation between responses at each time point the correlation within separate responses over time and the cross correlation between different responses at different times in addition the estimated covariance matrix should satisfy the positive definiteness condition and it may be heteroscedastic however in practice the structure of the covariance matrix is assumed to be homoscedastic and highly parsimonious such as exchangeable or autoregressive with order one these assumptions are too strong and result in inefficient estimates of the effects of covariates several studies have been conducted to solve these restrictions using modified cholesky decomposition mcd and linear covariance models however modeling the correlation between responses at each time point is not easy because there is no natural ordering of the responses in this paper we use mcd and hypersphere decomposition to model the complex correlation structures for multivariate longitudinal data we observe that the estimated covariance matrix using the decompositions is positive definite and can be heteroscedastic and that it is also interpretable the proposed methods are illustrated using data from a nonalcoholic fatty liver disease study

frechet differentiability of sp norms abstract one of the long standing questions in the theory of schatten von neumann ideals of compact operators is whether their norms have the same differentiability properties as the norms of their commutative counterparts we answer this question in the affirmative a key technical observation underlying our proof is a discovery of connection between this question and recent affirmative resolution of l s koplienko s conjecture concerning existence of higher order spectral shift functions

detection and discrimination of frequency glides as a function of direction duration frequency span and center frequency the study investigated the ability to detect and discriminate frequency glides under a variety of experimental conditions the subjects distinguished between a comparison signal that either was level in frequency or was swept across a fixed frequency span and a target signal that changed more in frequency than the comparison signal tone durations were 50 and 400 ms nominal center frequencies were 0 5 2 and 6 khz actual center frequencies were varied randomly or roved over a range equal to 0 1 times the nominal center frequency up and down glides were used the transition span of the comparison signal was either 0 0 5 1 or 2 times the equivalent rectangular bandwidth of the auditory filter at the nominal center frequency discrimination thresholds were obtained for all combinations of center frequency direction and span overall thresholds expressed as hz erb varied little as a function of center frequency glide duration had no effect on discrimination the 50 ms down glides were more dif

consistent model selection and data driven smooth tests for longitudinal data in the estimating equations approach summary model selection for marginal regression analysis of longitudinal data is challenging owing to the presence of correlation and the difficulty of specifying the full likelihood particularly for correlated categorical data the paper introduces a novel bayesian information criterion type model selection procedure based on the quadratic inference function which does not require the full likelihood or quasi likelihood with probability approaching 1 the criterion selects the most parsimonious correct model although a working correlation matrix is assumed there is no need to estimate the nuisance parameters in the working correlation matrix moreover the model selection procedure is robust against the misspecification of the working correlation matrix the criterion proposed can also be used to construct a data driven neyman smooth test for checking the goodness of fit of a postulated model this test is especially useful and often yields much higher power in situations where the classical directional test behaves poorly the finite sample performance of the model selection and model checking procedures is demonstrated through monte carlo studies and analysis of a clinical trial data set

functions of unitary operators derivatives and trace formulas abstract let a be a bounded self adjoint operator on a separable hilbert space h and u 0 an arbitrary unitary operator we derive formulas for higher order derivatives of the function s e i s a u 0 for sufficiently smooth functions we compare our formulas with earlier results in this area and apply them to establish higher order analogs of trace formulas of krein koplienko neidhardt

nonparametric estimation of large covariance matrices of longitudinal data summary estimation of an unstructured covariance matrix is difficult because of its positivedefiniteness constraint this obstacle is removed by regressing each variable on its predecessors so that estimation of a covariance matrix is shown to be equivalent to that of estimating a sequence of varying coefficient and varying order regression models our framework is similar to the use of increasing order autoregressive models in approximating the covariance matrix or the spectrum of a stationary time series as an illustration we adopt fan zhang s 2000 two step estimation of functional linear models and propose nonparametric estimators of covariance matrices which are guaranteed to be positive definite for parsimony a suitable order for the sequence of auto regression models is found using penalised likelihood criteria like aic and bic some asymptotic results for the local polynomial estimators of components of a covariance matrix are established two longitudinal datasets are analysed to illustrate the methodology a simulation study reveals the advantage of the nonparametric covariance estimator over the sample covariance matrix for large covariance matrices

noncommutative riesz transforms a probabilistic approach for 2 p 0 in the classical setting this is certainly satisfied for the maps tt f f 1 t where t is a semigroup of measure preserving maps on a probability space in that case n l then the maps tt act on all noncommutative lp spaces lp n and in particular on the hilbert space l2 l2 n let a be the negative generator i e tt e ta for technical reasons we have to assume that there is a weakly dense algebra a n which lies in the domain of the unbounded operator a i e a a a the algebra a plays the role of smooth functions in the classical setting this assumption might be too restrictive for some classical applications but it is easily verified in the new noncommutative examples for elements x y a we may define the gradient form 2 x y a x y x a y a x y the correct definition of is the one given by sauvageot as a bilinear form with values in l1 again our arguments require stronger assumptions in fact in all our examples we will assume that the tt s are selfadjoint i e ttxy xtty and unital i e tt 1 1 then a is indeed a positive unbounded operator under these circumstances we can formulate p a meyer s problem is it true that x x 2 p c p a 1 2 x p 0 1 holds for all selfadjoint elements x a let us illustrate this question by considering the laplace operator a f f where i 2 x2 i then it is easily verified that

on the arazy conjecture concerning schur multipliers on schatten ideals abstract we confirm in the affirmative the conjecture made by arazy in 2 suppose that f is a continuously differentiable function on 1 1 such that f t c t t 1 1 for some constant c 0 and 0 let j j 1 l r for some 0 r if 0 p and 1 q satisfy 1 p r 1 q then the matrix f j f k j k j k 1 is a schur multiplier from the schatten class s q into s p we also consider a general version of this conjecture for arbitrary quasi banach ideals of compact operators

an estimating equation approach to dimension reduction for longitudinal data sufficient dimension reduction has been extensively explored in the context of independent and identically distributed data in this article we generalize sufficient dimension reduction to longitudinal data and propose an estimating equation approach to estimating the central mean subspace the proposed method accounts for the covariance structure within each subject and improves estimation efficiency when the covariance structure is correctly specified even if the covariance structure is misspecified our estimator remains consistent in addition our method relaxes distributional assumptions on the covariates and is doubly robust to determine the structural dimension of the central mean subspace we propose a bayesian type information criterion we show that the estimated structural dimension is consistent and that the estimated basis directions are root formula see text consistent asymptotically normal and locally efficient simulations and an analysis of the framingham heart study data confirm the effectiveness of our approach

discrimination of nonlinear frequency glides discrimination thresholds for short duration nonlinear tone glides that differed in glide rate were measured in order to determine whether cues related to rate of frequency change alone were sufficient for discrimination thresholds for rising and falling nonlinear glides of 50 ms and 400 ms duration spanning three frequency excursions 0 5 1 and 2 erbs at three center frequencies 0 5 2 0 and 6 0khz were measured results showed that glide discrimination was possible when duration and initial and final frequencies were identical thresholds were of a different order to those found in previous studies using linear frequency glides where endpoint frequency or duration information is available as added cues the pattern of results was suggestive of a mechanism sensitive to spectral changes in time thresholds increased as the rate of transition span increased particularly above spans of 1 erb the weber fraction associated with these changes was 0 6 0 7 overall the results were consistent with an