corrected confidence bands for functional data using principal components functional principal components fpc analysis is widely used to decompose and express functional observations curve estimates implicitly condition on basis functions and other quantities derived from fpc decompositions however these objects are unknown in practice in this article we propose a method for obtaining correct curve estimates by accounting for uncertainty in fpc decompositions additionally pointwise and simultaneous confidence intervals that account for both model and decomposition based variability are constructed standard mixed model representations of functional expansions are used to construct curve estimates and variances conditional on a specific decomposition iterated expectation and variance formulas combine model based conditional estimates across the distribution of decompositions a bootstrap procedure is implemented to understand the uncertainty in principal component decomposition quantities our method compares favorably to competing approaches in simulation studies that include both densely and sparsely observed functions we apply our method to sparse observations of cd4 cell counts and to dense white matter tract profiles code for the analyses and simulations is publicly available and our method is implemented in the r package refund on cran

dimension reduction in a semiparametric regression model with errors in covariates we consider a semiparametric estimation method for general regression models when some of the predictors are measured with error the technique relies on a kernel regression of the true covariate on all the observed covariates and surrogates this requires a nonparametric regression in as many dimensions as there are covariates and surrogates the usual theory copes with such higher dimensional problems by using higher order kernels but this is unrealistic for most problems we show that the usual theory is essentially as good as one can do with this technique instead of regression with higher order kernels we propose the use of dimension reduction techniques we assume that the true covariate depends only on a linear combination of the observed covariates and surrogates if this linear combination were known we could apply the one dimensional versions of the semiparametric problem for which standard kernels are applicable we show that if one can estimate the linear directions at the root n rate then asymptotically the resulting estimator of the parameters in the main regression model behaves as if the linear combination were known simulations lend some credence to the asymptotic results

nonparametric inferences for additive models additive models with backfitting algorithms are popular multivariate nonparametric fitting techniques however the inferences of the models have not been very well developed due partially to the complexity of the backfitting estimators there are few tools available to answer some important and frequently asked questions such as whether a specific additive component is significant or admits a certain parametric form in an attempt to address these issues we extend the generalized likelihood ratio glr tests to additive models using the backfitting estimator we demonstrate that under the null models the newly proposed glr statistics follow asymptotically rescaled chi squared distributions with the scaling constants and the degrees of freedom independent of the nuisance parameters this demonstrates that the wilks phenomenon continues to hold under a variety of smoothing techniques and more relaxed models with unspecified error distributions we further prove that the glr tests are asymptotically o

entropy based importance measure for uncertain model inputs in the information theory entropy is a measure of the uncertainty associated with a random variable to identify the important variables of a system subjected to many random variables a novel entropy based importance measure for random variables is proposed in this paper the method evaluates the effect of a random variable on the output responses by calculating the change of the entropy value this technique focuses on the influence of input uncertainty on the entropy of the output responses and can be easily extended to the case taking the correlations among the random variables into consideration besides the effect of a random variable within any partial region of interest can also be evaluated by the proposed global sensitivity indicator the mathematical properties of the proposed importance measure are investigated and proven in detail three simple numerical examples are employed to demonstrate the applicability of the new importance measure and then a probabilistic risk assessment model is us

joint modelling of paired sparse functional data using principal components we propose a modelling framework to study the relationship between two paired longitudinally observed variables the data for each variable are viewed as smooth curves measured at discrete time points plus random errors while the curves for each variable are summarized using a few important principal components the association of the two longitudinal variables is modelled through the association of the principal component scores we use penalized splines to model the mean curves and the principal component curves and cast the proposed model into a mixed effects model framework for model fitting prediction and inference the proposed method can be applied in the difficult case in which the measurement times are irregular and sparse and may differ widely across individuals use of functional principal components enhances model interpretation and improves statistical and numerical stability of the parameter estimates

compactness conditions in the geometric theory of banach spaces in the last few years a lot of papers have appeared devoted to the branch of geometric theory of banach spaces involving compactness conditions the notions being fundamental in the classical geometry of banach spaces such as strict convexity uniform convexity smoothness and uniform smoothness 141 have been translated in terms of a measure of noncompactness by huff 13 goebel and sekowski 111 sekowski and stachura 25 rolewicz 22 231 montesinos 18 and the present author 2 31 these new concepts turn out to be useful tools both in the geometric theory of banach spaces and for many applications cf 3 6 10 11 13 18 20 21 241 the purpose of this paper is to discuss several properties connected with the concepts mentioned above firstly we define the modulus of near convexity which corresponds to the concept introduced by huff 13 it will be shown that this modulus is in a certain sense the inverse function to the modulus of noncompact convexity introduced by goebel and sekowski 111 secondly using the concept of a nearly uniformly smooth banach space 3 we introduce the notion of a modulus of near smoothness and we describe a certain relationship between this modulus and the modulus of near convexity this relationship is an analogue of the classical lindenstrauss dual formula 161 based on this relationship and on properties of the mentioned moduli we can calculate several formulas for the moduli of noncompact convexity near convexity and near smoothness in some classical banach spaces we also extend the classical results due to day 7 81 concerning the classification of the product spaces namely we shall show that some product spaces which are not uniformly convex are nearly uniformly convex or have the so called drop property similar results concerning the concept of smoothness will be also obtained the results of this paper form a continuation of the study from a previous article 3

on the nonparametric estimation of covariance functions we describe kernel methods for estimating the covariance function of a stationary stochastic process and show how to ensure that the estimator has the positive semidefiniteness property from a practical viewpoint our method is significant because it does not demand a parametric model for covariance from a technical angle our results exhibit a striking departure from those in more familiar cases of kernel estimation for example in the context of covariance estimation kernel estimators can have the same convergence rates as maximum likelihood estimators and can have exceptionally fast convergence rates when employed to estimate variance

the fixed point property for mappings admitting a center abstract we introduce a class of nonlinear continuous mappings in banach spaces which allow us to characterize the banach spaces without noncompact flat parts in their spheres as those that have the fixed point property for this type of mapping later on we give an application to the existence of zeroes for certain kinds of accretive operators

locally adaptive bandwidth choice for kernel regression estimators abstract kernel estimators with a global bandwidth are commonly used to estimate regression functions on the other hand it is obvious that the choice of a local bandwidth can lead to better results because a larger class of kernel estimators is available evidently this may in turn affect variability the optimal bandwidths depend essentially on the regression function itself and on the residual variance and it is desirable to estimate them from the data in this article a local bandwidth estimator is studied a comparison with its global bandwidth equivalent is performed both in theory and in simulations as the main result it is shown that the possible gain in mean integrated squared error of the resulting regression estimator must be paid for by a larger variability of the estimator this may lead to worse results if the sample size is small an algorithm has been devised that puts special weight on stability aspects our simulation study shows that improvements over a global bandwidth estimator

some geometric properties related to smoothness of banach spaces abstract in this paper we introduce a new class of banach spaces locally k nearly uniformly smooth spaces which is the dual notion to locally k nearly uniformly convex spaces introduced by denka kutzarova the relationships between locally k nearly uniformly smooth spaces locally k uniformly smooth banach spaces and locally fully k smooth banach spaces are discussed we also introduce the two new smooth properties s k i and s k ii which are the extensions of smooth properties s i and s ii that were introduced by suyalatu and discuss the relations between s k i s k ii and other smooth properties at last the locally fully k smoothness compact locally fully k smoothness smooth properties s k i and s k ii in product space l p x i are discussed

sequential subspace reliability method with univariate revolving integration a new computational framework called the sequential subspace reliability method ssrm is presented this method decomposes the multidimensional random space into multiple two dimensional subspaces in this manner ssrm is able to approximate bivariate interaction effects when the reliability estimate contribution is calculated subspace by subspace the final assessment is updated in a progressive manner the iterative history of sequential reliability assessment can be used to understand the complexity and convergence behavior of the limit state function of interest in a decision making situation the flexibility of the proposed ssrm to provide iterative updates on reliability estimation becomes especially valuable in dealing with large scale and complex problems under the constraints of limited time and resources to calculate the individual subspace contributions a novel univariate revolving integration uri method is proposed the uri method takes advantage of the axisymmetric nature of a joint pr

uniform convergence rates for nonparametric regression and principal component analysis in functional longitudinal data we consider nonparametric estimation of the mean and covariance functions for functional longitudinal data strong uniform convergence rates are developed for estimators that are local linear smoothers our results are obtained in a unified framework in which the number of observations within each curve cluster can be of any rate relative to the sample size we show that the convergence rates for the procedures depend on both the number of sample curves and the number of observations on each curve for sparse functional data these rates are equivalent to the optimal rates in nonparametric regression for dense functional data root n rates of convergence can be achieved with proper choices of bandwidths we further derive almost sure rates of convergence for principal component analysis using the estimated covariance function the results are illustrated with simulation studies

nested hierarchical functional data modeling and inference for the analysis of functional plant phenotypes abstractin a plant science root image study the process of seedling roots bending in response to gravity is recorded using digital cameras and the bending rates are modeled as functional plant phenotype data the functional phenotypes are collected from seeds representing a large variety of genotypes and have a three level nested hierarchical structure with seeds nested in groups nested in genotypes the seeds are imaged on different days of the lunar cycle and an important scientific question is whether there are lunar effects on root bending we allow the mean function of the bending rate to depend on the lunar day and model the phenotypic variation between genotypes groups of seeds imaged together and individual seeds by hierarchical functional random effects we estimate the covariance functions of the functional random effects by a fast penalized tensor product spline approach perform multi level functional principal component analysis fpca using the best linear unbiased predictor of the pri

functional mixed effects models in this article a new class of functional models in which smoothing splines are used to model fixed effects as well as random effects is introduced the linear mixed effects models are extended to nonparametric mixed effects models by introducing functional random effects which are modeled as realizations of zero mean stochastic processes the fixed functional effects and the random functional effects are modeled in the same functional space which guarantee the population average and subject specific curves have the same smoothness property these models inherit the flexibility of the linear mixed effects models in handling complex designs and correlation structures can include continuous covariates as well as dummy factors in both the fixed or random design matrices and include the nested curves models as special cases two estimation procedures are proposed the first estimation procedure exploits the connection between linear mixed effects models and smoothing splines and can be fitted using existing software the second procedure is a sequential estimation procedure using kalman filtering this algorithm avoids inversion of large dimensional matrices and therefore can be applied to large data sets a generalized maximum likelihood gml ratio test is proposed for inference and model selection an application to comparison of cortisol profiles is used as an illustration

generalized quasi likelihood ratio tests for semiparametric analysis of covariance models in longitudinal data abstractwe model generalized longitudinal data from multiple treatment groups by a class of semiparametric analysis of covariance models which take into account the parametric effects of time dependent covariates and the nonparametric time effects in these models the treatment effects are represented by nonparametric functions of time and we propose a generalized quasi likelihood ratio test procedure to test if these functions are identical our estimation procedure is based on profile estimating equations combined with local linear smoothers we find that the much celebrated wilks phenomenon which is well established for independent data still holds for longitudinal data if a working independence correlation structure is assumed in the test statistic however this property does not hold in general especially when the working variance function is misspecified our empirical study also shows that incorporating correlation into the test statistic does not necessarily improve the power of the test the

sequential subspace robustness assessment and sensitivity analysis to avoid an overconservative design and to ensure desired performance in an optimal way the product quality and robustness are considered in terms of the product performance mean and variance in this paper to facilitate robust design exploration under uncertainty a new sequential subspace robustness assessment method is presented to assess not only the mean and variance of performance but also their sensitivities with respect to design parameters the proposed method is based on the computational framework that integrates the univariate revolving integration and surrogate modeling of univariate integral functions the proposed framework enables consideration of bivariate interaction effects approximately by the aggregation of multiple revolving integration terms in a partial set of bivariate subspaces it is found that the proposed method provides better accuracy with comparable computational cost in assessing the statistical moments and sensitivities of product performance than existing methods suc

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

modeling partly conditional means with longitudinal data abstract we propose a general modeling approach to longitudinal data that is a hybrid of the marginal regression models of zeger and liang and of the classical transition models such as used in time series analyses rather than conditioning at time t only on covariate values as is typical with the marginal approach or on the entire history of the process up to t as is typical with the transition model approach we suggest models that condition on a subset of the process history estimation proceeds using generalized estimating equation methodology but with the restriction that the working covariance matrix is diagonal the proposed regression models share common features with cox regression models for failure time data in that they are composed of a nuisance baseline function of time and a simple parametric function of the covariates two illustrative examples are presented

three part joint modeling methods for complex functional data mixed with zero and one inflated proportions and zero inflated continuous outcomes with skewness we take a functional data approach to longitudinal studies with complex bivariate outcomes this work is motivated by data from a physical activity study that measured 2 responses over time in 5 minute intervals one response is the proportion of time active in each interval a continuous proportions with excess zeros and ones the other response energy expenditure rate in the interval is a continuous variable with excess zeros and skewness this outcome is complex because there are 3 possible activity patterns in each interval inactive partially active and completely active and those patterns which are observed induce both nonrandom and random associations between the responses more specifically the inactive pattern requires a zero value in both the proportion for active behavior and the energy expenditure rate a partially active pattern means that the proportion of activity is strictly between zero and one and that the energy expenditure rate is greater than zero and likely to be moderate and the completely active pattern means that the proportion of activity is exactly one and the energy expenditure rate is greater than zero and likely to be higher to address these challenges we propose a 3 part functional data joint modeling approach the first part is a continuation ratio model to reorder the ordinal valued 3 activity patterns the second part models the proportions when they are in interval 0 1 the last component specifies the skewed continuous energy expenditure rate with box cox transformations when they are greater than zero in this 3 part model the regression structures are specified as smooth curves measured at various time points with random effects that have a correlation structure the smoothed random curves for each variable are summarized using a few important principal components and the association of the 3 longitudinal components is modeled through the association of the principal component scores the difficulties in handling the ordinal and proportional variables are addressed using a quasi likelihood type approximation we develop an efficient algorithm to fit the model that also involves the selection of the number of principal components the method is applied to physical activity data and is evaluated empirically by a simulation study

nonnegative decomposition of functional count data we present a novel decomposition of nonnegative functional count data that draws on concepts from nonnegative matrix factorization our decomposition which we refer to as narfd nonnegative and regularized function decomposition enables the study of patterns in variation across subjects in a highly interpretable manner prototypic modes of variation are estimated directly on the observed scale of the data are local and are transparently added together to reconstruct observed functions this contrasts with generalized functional principal component analysis an alternative approach that estimates functional principal components on a transformed scale produces components that typically vary across the entire functional domain and reconstructs observations using complex patterns of cancellation and multiplication of functional principal components narfd is implemented using an alternating minimization algorithm and we evaluate our approach in simulations we apply narfd to an accelerometer dataset comprising observations of physical activity for healthy older americans

first order saddlepoint approximation for reliability analysis in the approximation methods of reliability analysis nonnormal random variables are transformed into equivalent standard normal random variables this transformation tends to increase the nonlinearity of a limit state function and hence results in less accurate reliability approximation the first order saddlepoint approximation for reliability analysis is proposed to improve the accuracy of reliability analysis by the approximation of a limit state function at the most likelihood point in the original random space and employment of the accurate saddlepoint approximation the proposed method reduces the chance of an increase in the nonlinearity of the limit state function this approach generates more accurate reliability approximation than the first order reliability method without an increase in the computational effort the effectiveness of the proposed method is demonstrated with two examples and is compared with the first and second order reliability methods

nonparametric estimation of correlation functions in longitudinal and spatial data with application to colon carcinogenesis experiments in longitudinal and spatial studies observations often demonstrate strong correlations that are stationary in time or distance lags and the times or locations of these data being sampled may not be homogeneous we propose a nonparametric estimator of the correlation function in such data using kernel methods we develop a pointwise asymptotic normal distribution for the proposed estimator when the number of subjects is fixed and the number of vectors or functions within each subject goes to infinity based on the asymptotic theory we propose a weighted block bootstrapping method for making inferences about the correlation function where the weights account for the inhomogeneity of the distribution of the times or locations the method is applied to a data set from a colon carcinogenesis study in which colonic crypts were sampled from a piece of colon segment from each of the 12 rats in the experiment and the expression level of p27 an important cell cycle protein was then measured for each cell within the sampled crypts a simulation study is also provided to illustrate the numerical performance of the proposed method

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

wavelet based functional mixed models increasingly scientific studies yield functional data in which the ideal units of observation are curves and the observed data consist of sets of curves that are sampled on a fine grid we present new methodology that generalizes the linear mixed model to the functional mixed model framework with model fitting done by using a bayesian wavelet based approach this method is flexible allowing functions of arbitrary form and the full range of fixed effects structures and between curve covariance structures that are available in the mixed model framework it yields nonparametric estimates of the fixed and random effects functions as well as the various between curve and within curve covariance matrices the functional fixed effects are adaptively regularized as a result of the non linear shrinkage prior that is imposed on the fixed effects wavelet coefficients and the random effect functions experience a form of adaptive regularization because of the separately estimated variance components for each wavelet coefficient because we have posterior samples for all model quantities we can perform pointwise or joint bayesian inference or prediction on the quantities of the model the adaptiveness of the method makes it especially appropriate for modelling irregular functional data that are characterized by numerous local features like peaks

semiparametric models for longitudinal data with application to cd4 cell numbers in hiv seroconverters summary the paper describes a semiparametric model for longitudinal data which is illustrated by its application to data on the time evolution of cd4 cell numbers in hiv seroconverters the essential ingredients of the model are a parametric linear model for covariate adjustment a nonparametric estimation of a smooth time trend serial correlation between measurements on an individual subject and random measurement error a back fitting algorithm is used in conjunction with a cross validation prescription to fit the model a notable feature in the application is that the onset of hiv infection is associated with a sudden drop in cd4 cells followed by a longer term slower decay the model is also used to estimate an individual s curve by combining his data with the population curve shrinkage toward the population mean trajectory is controlled in a natural way by the estimated covariance structure of the data

registration for exponential family functional data we consider the problem of aligning curves from exponential family distributions the approach is based on the combination of alignment and functional principal components analysis and is facilitated by recent extensions of fpca to non gaussian settings our work is motivated by the study of physical activity using accelerometers wearable devices that provide around the clock monitoring of activity and produce non gaussian measurements we apply the proposed methods to activity counts using a poisson distribution and to a binary active vs inactive indicator using a binomial distribution after alignment the trajectories show clear peaks of activity in the morning and afternoon with a dip in the middle of the day

marginal nonparametric kernel regression accounting for within subject correlation there has been substantial recent interest in non and semiparametric methods for longitudinal or clustered data with dependence within clusters it has been shown rather inexplicably that when standard kernel smoothing methods are used in a natural way higher efficiency is obtained by assuming independence than by using the true correlation structure it is shown here that this result is a natural consequence of how standard kernel methods incorporate the within subject correlation in the asymptotic setting considered where the cluster sizes are fixed and the cluster number increases in this paper an alternative kernel smoothing method is proposed unlike the standard methods the smallest variance of the new estimator is achieved when the true correlation is assumed asymptotically the variance of the proposed method is uniformly smaller than that of the most efficient working independence approach a small simulation study shows that significant improvement is obtained for finite samples

semiparametric regression for clustered data using generalized estimating equations we consider estimation in a semiparametric generalized linear model for clustered data using estimating equations our results apply to the case where the number of observations per cluster is finite whereas the number of clusters is large the mean of the outcome variable is of the form g xt t where g is a link function x and t are covariates is an unknown parameter vector and t is an unknown smooth function kernel estimating equations proposed previously in the literature are used to estimate the infinite dimensional nonparametric function t and a profile based estimating equation is used to estimate the finite dimensional parameter vector we show that for clustered data this conventional profile kernel method often fails to yield a n consistent estimator of along with appropriate inference unless working independence is assumed or t is artificially undersmoothed in which case asymptotic inference is possible to gain insight into these results we derive the se

structural reliability optimization using an efficient safety index calculation procedure the objective of this paper is to conduct reliability based structural optimization in a multidisciplinary environment an efficient reliability analysis is developed by expanding the limit functions in terms of intermediate design variables the design constraints are approximated using multivariate splines in searching for the optimum the reduction in computational cost realized in safety index calculation and optimization are demonstrated through several structural problems this paper presents safety index computation analytical sensitivity analysis of reliability constraints and optimization using truss frame and plate examples

general saddlepoint approximations of marginal densities and tail probabilities saddlepoint approximations of marginal densities and tail probabilities of general nonlinear statistics are derived these are based on the expansion of the statistic up to the second order their accuracy is shown in a variety of examples including logit and probit models and rank estimators for regression

analysis of longitudinal data with semiparametric estimation of covariance function improving efficiency for regression coefficients and predicting trajectories of individuals are two important aspects in the analysis of longitudinal data both involve estimation of the covariance function yet challenges arise in estimating the covariance function of longitudinal data collected at irregular time points a class of semiparametric models for the covariance function by that imposes a parametric correlation structure while allowing a nonparametric variance function is proposed a kernel estimator for estimating the nonparametric variance function is developed two methods for estimating parameters in the correlation structure a quasi likelihood approach and a minimum generalized variance method are proposed a semiparametric varying coefficient partially linear model for longitudinal data is introduced and an estimation procedure for model coefficients using a profile weighted least squares approach is proposed sampling properties of the proposed estimation procedures are studied and asy

properties of principal component methods for functional and longitudinal data analysis the use of principal component methods to analyze functional data is appropriate in a wide range of different settings in studies of functional data analysis it has often been assumed that a sample of random functions is observed precisely in the continuum and without noise while this has been the traditional setting for functional data analysis in the context of longitudinal data analysis a random function typically represents a patient or subject who is observed at only a small number of randomly distributed points with nonnegligible measurement error nevertheless essentially the same methods can be used in both these cases as well as in the vast number of settings that lie between them how is performance affected by the sampling plan in this paper we answer that question we show that if there is a sample of n functions or subjects then estimation of eigenvalues is a semiparametric problem with root n consistent estimators even if only a few observations are made of each function and if each observation is encumbered by noise however estimation of eigenfunctions becomes a nonparametric problem when observations are sparse the optimal convergence rates in this case are those which pertain to more familiar function estimation settings we also describe the effects of sampling at regularly spaced points as opposed to random points in particular it is shown that there are often advantages in sampling randomly however even in the case of noisy data there is a threshold sampling rate depending on the number of functions treated above which the rate of sampling either randomly or regularly has negligible impact on estimator performance no matter whether eigenfunctions or eigenvectors are being estimated

efficient semiparametric regression for longitudinal data with nonparametric covariance estimation summary for longitudinal data when the within subject covariance is misspecified the semiparametric regression estimator may be inefficient we propose a method that combines the efficient semiparametric estimator with nonparametric covariance estimation and is robust against misspecification of covariance models we show that kernel covariance estimation provides uniformly consistent estimators for the within subject covariance matrices and the semiparametric profile estimator with substituted nonparametric covariance is still semiparametrically efficient the finite sample performance of the proposed estimator is illustrated by simulation in an application to cd4 count data from an aids clinical trial we extend the proposed method to a functional analysis of the covariance model

a semiparametric change point regression model for longitudinal observations many longitudinal studies involve relating an outcome process to a set of possibly time varying covariates giving rise to the usual regression models for longitudinal data when the purpose of the study is to investigate the covariate effects when experimental environment undergoes abrupt changes or to locate the periods with different levels of covariate effects a simple and easy to interpret approach is to introduce change points in regression coefficients in this connection we propose a semiparametric change point regression model in which the error process stochastic component is nonparametric and the baseline mean function functional part is completely unspecified the observation times are allowed to be subject specific and the number locations and magnitudes of change points are unknown and need to be estimated we further develop an estimation procedure that combines the recent advance in semiparametric analysis based on counting process argument and multiple change points inference an

empirical bias bandwidths for local polynomial nonparametric regression and density estimation abstract a data based local bandwidth selector is proposed for nonparametric regression by local fitting of polynomials the estimator called the empirical bias bandwidth selector ebbs is rather simple and easily allows multivariate predictor variables and estimation of any order derivative of the regression function ebbs minimizes an estimate of mean squared error consisting of a squared bias term plus a variance term the variance term used is exact not asymptotic though it involves the conditional variance of the response given the predictors that must be estimated the bias term is estimated empirically not from an asymptotic expression thus ebbs is similar to the double smoothing approach of hardle hall and marron and a local bandwidth selector of schucany but is developed here for a far wider class of estimation problems than what those authors considered ebbs is tested on simulated data and its performance seems quite satisfactory local polynomial smoothing of a histogram is a high

modelling sparse generalized longitudinal observations with latent gaussian processes summary in longitudinal data analysis one frequently encounters non gaussian data that are repeatedly collected for a sample of individuals over time the repeated observations could be binomial poisson or of another discrete type or could be continuous the timings of the repeated measurements are often sparse and irregular we introduce a latent gaussian process model for such data establishing a connection to functional data analysis the functional methods proposed are non parametric and computationally straightforward as they do not involve a likelihood we develop functional principal components analysis for this situation and demonstrate the prediction of individual trajectories from sparse observations this method can handle missing data and leads to predictions of the functional principal component scores which serve as random effects in this model these scores can then be used for further statistical analysis such as inference regression discriminant analysis or clustering we illustrate these non parametric methods with longitudinal data on primary biliary cirrhosis and show in simulations that they are competitive in comparisons with generalized estimating equations and generalized linear mixed models

efficient semiparametric marginal estimation for longitudinal clustered data we consider marginal generalized semiparametric partially linear models for clustered data lin and carroll derived the semiparametric efficient score function for this problem in the multivariate gaussian case but they were unable to construct a semiparametric efficient estimator that actually achieved the semiparametric information bound here we propose such an estimator and generalize the work to marginal generalized partially linear models we investigate asymptotic relative efficiencies of the estimators that ignore the within cluster correlation structure either in nonparametric curve estimation or throughout we evaluate the finite sample performance of these estimators through simulations and illustrate it using a longitudinal cd4 cell count dataset both theoretical and numerical results indicate that properly taking into account the within subject correlation among the responses can substantially improve efficiency

some tools for functional data analysis multivariate data analysis permits the study of observations which are finite sets of numbers but modern data collection situations can involve data or the processes giving rise to them which are functions functional data analysis involves infinite dimensional processes and or data the paper shows how the theory of l splines can support generalizations of linear modelling and principal components analysis to samples drawn from random functions spline smoothing rests on a partition of a function space into two orthogonal subspaces one of which contains the obvious or structural components of variation among a set of observed functions and the other of which contains residual components this partitioning is achieved through the use of a linear differential operator and we show how the theory of polynomial splines can be applied more generally with an arbitrary operator and associated boundary constraints these data analysis tools are illustrated by a study of variation in temperature precipitation patterns among some canadian weather stations

additive extensions to generalized estimating equation methods the methods underlying vector generalized additive models are extended to provide additive extensions to the generalized estimating equations approaches to multivariate regression problems of liang and zeger and the subsequent literature the methods are illustrated by two examples using correlated binary data concerning the presence or absence of several plant species at the same geographical site and chest pain in male workers from a workforce study with minor modification the extensions are shown to apply to longitudinal data

penalized spline models for functional principal component analysis summary we propose an iterative estimation procedure for performing functional principal component analysis the procedure aims at functional or longitudinal data where the repeated measurements from the same subject are correlated an increasingly popular smoothing approach penalized spline regression is used to represent the mean function this allows straightforward incorporation of covariates and simple implementation of approximate inference procedures for coefficients for the handling of the within subject correlation we develop an iterative procedure which reduces the dependence between the repeated measurements that are made for the same subject the resulting data after iteration are theoretically shown to be asymptotically equivalent in probability to a set of independent data this suggests that the general theory of penalized spline regression that has been developed for independent data can also be applied to functional data the effectiveness of the proposed procedure is demonstrated via a simulation study and an application to yeast cell cycle gene expression data

bootstrap and wild bootstrap for high dimensional linear models in this paper two bootstrap procedures are considered for the estimation of the distribution of linear contrasts and of f test statistics in high dimensional linear models an asymptotic approach will be chosen where the dimension p of the model may increase for sample size n rightarrow infty the range of validity will be compared for the normal approximation and for the bootstrap procedures furthermore it will be argued that the rates of convergence are different for the bootstrap procedures in this asymptotic framework this is in contrast to the usual asymptotic approach where p is fixed

generalized likelihood ratio statistics and wilks phenomenon likelihood ratio theory has had tremendous success in parametric inference due to the fundamental theory of wilks yet there is no general applicable approach for nonparametric inferences based on function estimation maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation setting even if they exist they are hard to find and can not be optimal as shown in this paper we introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics a new s wilks phenomenon is unveiled we demonstrate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are asymptotically distribution free and follow chi 2 distributions under null hypotheses for a number of useful hypotheses and a variety of useful models including gaussian white noise models nonparametric regression models varying coefficient models and generalized varying coefficient models we further demonstrate that generalized likelihood ratio statistics are asymptotically optimal in the sense that they achieve optimal rates of convergence given by ingster they can even be adaptively optimal in the sense of spokoiny by using a simple choice of adaptive smoothing parameter our work indicates that the generalized likelihood ratio statistics are indeed general and powerful for nonparametric testing problems based on function estimation

reliability sensitivity analysis with random and interval variables in reliability analysis and reliability based design sensitivity analysis identifies the relationship between the change in reliability and the change in the characteristics of uncertain variables sensitivity analysis is also used to identify the most significant uncertain variables that have the highest contributions to reliability most of the current sensitivity analysis methods are applicable for only random variables in many engineering applications however some of uncertain variables are intervals in this work a sensitivity analysis method is proposed for the mixture of random and interval variables six sensitivity indices are defined for the sensitivity of the average reliability and reliability bounds with respect to the averages and widths of intervals as well as with respect to the distribution parameters of random variables the equations of these sensitivity indices are derived based on the first order reliability method form the proposed reliability sensitivity analysis is a byproduct of form without any extra function calls after reliability is found once form is performed the sensitivity information is obtained automatically two examples are used for demonstration copyright 2009 john wiley sons ltd

semiparametric estimation of covariance matrixes for longitudinal data estimation of longitudinal data covariance structure poses significant challenges because the data usually are collected at irregular time points a viable semiparametric model for covariance matrixes has been proposed that allows one to estimate the variance function nonparametrically and to estimate the correlation function parametrically by aggregating information from irregular and sparse data points within each subject but the asymptotic properties of the quasi maximum likelihood estimator qmle of parameters in the covariance model are largely unknown we address this problem in the context of more general models for the conditional mean function including parametric nonparametric or semiparametric we also consider the possibility of rough mean regression function and introduce the difference based method to reduce biases in the context of varying coefficient partially linear mean regression models this provides a more robust estimator of the covariance function under a wider range of situati

semiparametric estimation in general repeated measures problems summary the paper considers a wide class of semiparametric problems with a parametric part for some covariate effects and repeated evaluations of a nonparametric function special cases in our approach include marginal models for longitudinal or clustered data conditional logistic regression for matched case control studies multivariate measurement error models generalized linear mixed models with a semiparametric component and many others we propose profile kernel and backfitting estimation methods for these problems derive their asymptotic distributions and show that in likelihood problems the methods are semiparametric efficient although generally not true it transpires that with our methods profiling and backfitting are asymptotically equivalent we also consider pseudolikelihood methods where some nuisance parameters are estimated from a different algorithm the methods proposed are evaluated by using simulation studies and applied to the kenya haemoglobin data

modelling and smoothing parameter estimation with multiple quadratic penalties summary penalized likelihood methods provide a range of practical modelling tools including spline smoothing generalized additive models and variants of ridge regression selecting the correct weights for penalties is a critical part of using these methods and in the single penalty case the analyst has several well founded techniques to choose from however many modelling problems suggest a formulation employing multiple penalties and here general methodology is lacking a wide family of models with multiple penalties can be fitted to data by iterative solution of the generalized ridge regression problem minimise kw 1 2 xp a y k 2 o p m i 1 iip 0 s i p p is a parameter vector x a design matrix si a non negative definite coefficient matrix defining the i th penalty with associated smoothing parameter ii w a diagonal weight matrix y a vector of data or pseudodata and o an overall smoothing parameter included for reasons of computational efficiency this paper shows how smoothing parameter selection can be performed efficiently by applying generalized cross validation to this problem and how this allows non linear generalized linear and linear models to be fitted using multiple penalties substantially increasing the scope of penalized modelling methods examples of non linear modelling generalized additive modelling and anisotropic smoothing are given

generalized linear models with random effects a gibbs sampling approach abstract generalized linear models have unified the approach to regression for a wide variety of discrete continuous and censored response variables that can be assumed to be independent across experimental units in applications such as longitudinal studies genetic studies of families and survey sampling observations may be obtained in clusters responses from the same cluster cannot be assumed to be independent with linear models correlation has been effectively modeled by assuming there are cluster specific random effects that derive from an underlying mixing distribution extensions of generalized linear models to include random effects has thus far been hampered by the need for numerical integration to evaluate likelihoods in this article we cast the generalized linear random effects model in a bayesian framework and use a monte carlo method the gibbs sampler to overcome the current computational limitations the resulting algorithm is flexible to easily accommodate changes in the number

weak and strong uniform consistency of kernel regression estimates we study the estimation of a regression function by the kernel method under mild conditions on the window the bandwidth and the underlying distribution of the bivariate observations x i y i we obtain the weak and strong uniform convergence rates on a bounded interval these results parallel those of silverman 1978 on density estimation and extend those of schuster and yakowitz 1979 and collomb 1979 on regression estimation

equivalent kernels of smoothing splines in nonparametric regression for clustered longitudinal data correlation we show that a smoothing spline estimator is asymptotically equivalent to a recently proposed seemingly unrelated kernel estimator of wang 2003 for any working covariance matrix we show that both estimators can be obtained iteratively by applying conventional kernel or spline smoothing to pseudo observations this result allows us to study the asymptotic properties of the smoothing spline estimator by deriving its asymptotic bias and variance we show that smoothing splines are consistent for an arbitrary working covariance and have the smallest variance when assuming the true covariance we further show that both the seemingly unrelated kernel estimator and the smoothing spline estimator are nonlocal unless working independence is assumed but have asymptotically negligible bias their finite sample performance is compared through simulations our results justify the use of efficient non local estimators such as smoothing splines for clustered longitudinal data

reliability analysis for multidisciplinary systems with random and interval variables tremendous efforts have been devoted to developing efficient approaches to reliability analysis for multidisciplinary systems most of the approaches are only capable of dealing with random variables modeled by probability distributions both random and interval variables however may exist in multidisciplinary systems their propagation through coupled subsystems make s reliability analysis computationally expensive in this work a unified reliability analysis framework is proposed to deal with both random and interval variables in multidisciplinary systems the framework is an extension of an existing unified uncertainty analysis framework for single disciplinary problems the new framework involves probabilistic analysis and interval analysis both probabilistic analysis and interval analysis are decoupled from each other and are performed sequentially the first order reliability method is used for probabilistic analysis three supporting algorithms are developed the effectiveness of the algorithms is demonstrated with a mathematical example and an engineering application

nonparametric estimation of covariance structure in longitudinal data summary in longitudinal studies the effect of various treatments over time is usually of prime interest however observations on the same subject are usually correlated and any analysis should account for the underlying covariance structure a nonparametric estimate of the covariance structure is useful either as a guide to the formulation of a parametric model or as the basis for formal inference without imposing parametric assumptions the sample covariance matrix provides such an estimate when the data consist of a short sequence of measurements at a common set of time points on each of many subjects but is impractical when the data are severely unbalanced or when the sequences of measurements on individual subjects are long relative to the number of subjects the variogram of residuals from a saturated model for the mean response has previously been suggested as a nonparametric estimator for covariance structure assuming stationarity in this paper we consider kernel weighted local linear regression smoothing of sample variogram ordinates and of squared residuals to provide a nonparametric estimator for the covariance structure without assuming stationarity the value of the estimator as a diagnostic tool is demonstrated in two applications one to a set of data concerning the blood pressure of newborn babies in an intensive care unit and the other to data on the time evolution of cd4 cell numbers in hiv seroconverters

multilevel functional clustering analysis in this article we investigate clustering methods for multilevel functional data which consist of repeated random functions observed for a large number of units e g genes at multiple subunits e g bacteria types to describe the within and between variability induced by the hierarchical structure in the data we take a multilevel functional principal component analysis mfpca approach we develop and compare a hard clustering method applied to the scores derived from the mfpca and a soft clustering method using an mfpca decomposition in a simulation study we assess the estimation accuracy of the clustering membership and the cluster patterns under a series of settings small versus moderate number of time points various noise levels and varying number of subunits per unit we demonstrate the applicability of the clustering analysis to a real data set consisting of expression profiles from genes activated by immunity system cells prevalent response patterns are identified by clustering the expression profiles using our multilevel clustering analysis

reduced rank mixed effects models for spatially correlated hierarchical functional data hierarchical functional data are widely seen in complex studies where subunits are nested within units which in turn are nested within treatment groups we propose a general framework of functional mixed effects model for such data within unit and within subunit variations are modeled through two separate sets of principal components the subunit level functions are allowed to be correlated penalized splines are used to model both the mean functions and the principal components functions where roughness penalties are used to regularize the spline fit an expectation maximization em algorithm is developed to fit the model while the specific covariance structure of the model is utilized for computational efficiency to avoid storage and inversion of large matrices our dimension reduction with principal components provides an effective solution to the difficult tasks of modeling the covariance kernel of a random function and modeling the correlation between functions the proposed methodology is illus

generalized multilevel function on scalar regression and principal component analysis this manuscript considers regression models for generalized multilevel functional responses functions are generalized in that they follow an exponential family distribution and multilevel in that they are clustered within groups or subjects this data structure is increasingly common across scientific domains and is exemplified by our motivating example in which binary curves indicating physical activity or inactivity are observed for nearly 600 subjects over 5 days we use a generalized linear model to incorporate scalar covariates into the mean structure and decompose subject specific and subject day specific deviations using multilevel functional principal components analysis thus functional fixed effects are estimated while accounting for within function and within subject correlations and major directions of variability within and between subjects are identified fixed effect coefficient functions and principal component basis functions are estimated using penalized splines model parameters are estimated in a bayesian framework using stan a programming language that implements a hamiltonian monte carlo sampler simulations designed to mimic the application have good estimation and inferential properties with reasonable computation times for moderate datasets in both cross sectional and multilevel scenarios code is publicly available in the application we identify effects of age and bmi on the time specific change in probability of being active over a 24 hour period in addition the principal components analysis identifies the patterns of activity that distinguish subjects and days within subjects

smoothing spline models for the analysis of nested and crossed samples of curves abstract we introduce a class of models for an additive decomposition of groups of curves stratified by crossed and nested factors generalizing smoothing splines to such samples by associating them with a corresponding mixed effects model the models are also useful for imputation of missing data and exploratory analysis of variance we prove that the best linear unbiased predictors blups from the extended mixed effects model correspond to solutions of a generalized penalized regression where smoothing parameters are directly related to variance components and we show that these solutions are natural cubic splines the model parameters are estimated using a highly efficient implementation of the em algorithm for restricted maximum likelihood reml estimation based on a preliminary eigenvector decomposition variability of computed estimates can be assessed with asymptotic techniques or with a novel hierarchical bootstrap resampling scheme for nested mixed effects models our methods are applied to me

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

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

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

nonparametric function estimation for clustered data when the predictor is measured without with error abstract we consider local polynomial kernel regression with a single covariate for clustered data using estimating equations we assume that at most m observations are available on each cluster in the case of random regressors with no measurement error in the predictor we show that it is generally the best strategy to ignore entirely the correlation structure within each cluster and instead pretend that all observations are independent in the further special case of longitudinal data on individuals with fixed common observation times we show that equivalent to the pooled data approach is the strategy of fitting separate nonparametric regressions at each observation time and constructing an optimal weighted average we also consider what happens when the predictor is measured with error using the simex approach to correct for measurement error we construct an asymptotic theory for both the pooled and the weighted average estimators surprisingly for the same amount of smoothing the weighted a

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

semiparametric stochastic mixed models for longitudinal data abstract we consider inference for a semiparametric stochastic mixed model for longitudinal data this model uses parametric fixed effects to represent the covariate effects and an arbitrary smooth function to model the time effect and accounts for the within subject correlation using random effects and a stationary or nonstationary stochastic process we derive maximum penalized likelihood estimators of the regression coefficients and the nonparametric function the resulting estimator of the nonparametric function is a smoothing spline we propose and compare frequentist inference and bayesian inference on these model components we use restricted maximum likelihood to estimate the smoothing parameter and the variance components simultaneously we show that estimation of all model components of interest can proceed by fitting a modified linear mixed model we illustrate the proposed method by analyzing a hormone dataset and evaluate its performance through simulations