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Abstract–In functional data analysis, data are commonly assumed to be smooth functions on a fixed interval of the real line. In this work, we introduce a comprehensive framework for the analysis of functional data, whose domain is a two-dimensional manifold and the domain itself is subject to variability from sample to sample. We formulate a statistical model for such data, here called functions on surfaces, which enables a joint representation of the geometric and functional aspects, and propose an associated estimation framework. We assess the validity of the framework by performing a simulation study and we finally apply it to the analysis of neuroimaging data of cortical thickness, acquired from the brains of different subjects, and thus lying on domains with different geometries. for this article are available online. [ABSTRACT FROM AUTHOR]

We propose a new class of filtering and smoothing methods for inference in high-dimensional, nonlinear, non-Gaussian, spatio-temporal state-space models. The main idea is to combine the ensemble Kalman filter and smoother, developed in the geophysics literature, with state-space algorithms from the statistics literature. Our algorithms address a variety of estimation scenarios, including online and off-line state and parameter estimation. We take a Bayesian perspective, for which the goal is to generate samples from the joint posterior distribution of states and parameters. The key benefit of our approach is the use of ensemble Kalman methods for dimension reduction, which allows inference for high-dimensional state vectors. We compare our methods to existing ones, including ensemble Kalman filters, particle filters, and particle MCMC. Using a real data example of cloud motion and data simulated under a number of nonlinear and non-Gaussian scenarios, we show that our approaches outperform these existing methods. for this article are available online. [ABSTRACT FROM AUTHOR]

Dialect variation is of considerable interest in linguistics and other social sciences. However, traditionally it has been studied using proxies (transcriptions) rather than acoustic recordings directly. We introduce novel statistical techniques to analyze geolocalized speech recordings and to explore the spatial variation of pronunciations continuously over the region of interest, as opposed to traditional isoglosses, which provide a discrete partition of the region. Data of this type require an explicit modeling of the variation in the mean and the covariance. Usual Euclidean metrics are not appropriate, and we therefore introduce the concept of d-covariance, which allows consistent estimation both in space and at individual locations. We then propose spatial smoothing for these objects which accounts for the possibly nonconvex geometry of the domain of interest. We apply the proposed method to data from the spoken part of the British National Corpus, deposited at the British Library, London, and we produce maps of the dialect variation over Great Britain. In addition, the methods allow for acoustic reconstruction across the domain of interest, allowing researchers to listen to the statistical analysis. for this article are available online. [ABSTRACT FROM AUTHOR]

Literature on change point analysis mostly requires a sudden change in the data distribution, either in a few parameters or the distribution as a whole. We are interested in the scenario, where the variance of data may make a significant jump while the mean changes in a smooth fashion. The motivation is a liver procurement experiment monitoring organ surface temperature. Blindly applying the existing methods to the example can yield erroneous change point estimates since the smoothly changing mean violates the sudden-change assumption. We propose a penalized weighted least-squares approach with an iterative estimation procedure that integrates variance change point detection and smooth mean function estimation. The procedure starts with a consistent initial mean estimate ignoring the variance heterogeneity. Given the variance components the mean function is estimated by smoothing splines as the minimizer of the penalized weighted least squares. Given the mean function, we propose a likelihood ratio test statistic for identifying the variance change point. The null distribution of the test statistic is derived together with the rates of convergence of all the parameter estimates. Simulations show excellent performance of the proposed method. Application analysis offers numerical support to non invasive organ viability assessment by surface temperature monitoring. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

We consider the estimation of mean and covariance functions for longitudinal and sparse functional data by using the full quasi-likelihood coupling a modification of the local kernel smoothing method. The proposed estimators are shown to be consistent, asymptotically normal, and semiparametrically efficient in terms of their linear functionals. Their superiority to the competitors is further illustrated numerically through simulation studies. The method is applied to analyze AIDS study and atmospheric study. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

We present false discovery rate (FDR) smoothing, an empirical-Bayes method for exploiting spatial structure in large multiple-testing problems. FDR smoothing automatically finds spatially localized regions of significant test statistics. It then relaxes the threshold of statistical significance within these regions, and tightens it elsewhere, in a manner that controls the overall false discovery rate at a given level. This results in increased power and cleaner spatial separation of signals from noise. The approach requires solving a nonstandard high-dimensional optimization problem, for which an efficient augmented-Lagrangian algorithm is presented. In simulation studies, FDR smoothing exhibits state-of-the-art performance at modest computational cost. In particular, it is shown to be far more robust than existing methods for spatially dependent multiple testing. We also apply the method to a dataset from an fMRI experiment on spatial working memory, where it detects patterns that are much more biologically plausible than those detected by standard FDR-controlling methods. All code for FDR smoothing is publicly available in Python and R (https://github.com/tansey/smoothfdr). Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

We present an offline, iterated particle filter to facilitate statistical inference in general state space hidden Markov models. Given amodel and a sequence of observations, the associated marginal likelihood L is central to likelihood-based inference for unknown statistical parameters. We define a class of "twisted" models: each member is specified by a sequence of positive functions ψ and has an associated ψ-auxiliary particle filter that provides unbiased estimates of L. We identify a sequence ψ* that is optimal in the sense that the ψ*-auxiliary particle filter's estimate of L has zero variance. In practical applications, ψ* is unknown so the ψ*-auxiliary particle filter cannot straightforwardly be implemented. We use an iterative scheme to approximateψ* and demonstrate empirically that the resulting iterated auxiliary particle filter significantly outperforms the bootstrap particle filter in challenging settings. Applications include parameter estimation using a particle Markov chain Monte Carlo algorithm. [ABSTRACT FROM AUTHOR]

An important problem in contemporary statistics is to understand the relationship among a large number of variables based on a dataset, usually withp, the number of the variables, much larger thann, the sample size. Recent efforts have focused on modeling static covariance matrices where pairwise covariances are considered invariant. In many real systems, however, these pairwise relations often change. To characterize the changing correlations in a high-dimensional system, we study a class of dynamic covariance models (DCMs) assumed to be sparse, and investigate for the first time a unified theory for understanding their nonasymptotic error rates and model selection properties. In particular, in the challenging high-dimensional regime, we highlight a new uniform consistency theory in which the sample size can be seen asn4/5when the bandwidth parameter is chosen ash∝n− 1/5for accounting for the dynamics. We show that this result holds uniformly over a range of the variable used for modeling the dynamics. The convergence rate bears the mark of the familiar bias-variance trade-off in the kernel smoothing literature. We illustrate the results with simulations and the analysis of a neuroimaging dataset. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

It has become commonplace to use complex computer models to predict outcomes in regions where data do not exist. Typically these models need to be calibrated and validated using some experimental data, which often consists of multiple correlated outcomes. In addition, some of the model parameters may be categorical in nature, such as a pointer variable to alternate models (or submodels) for some of the physics of the system. Here, we present a general approach for calibration in such situations where an emulator of the computationally demanding models and a discrepancy term from the model to reality are represented within a Bayesian smoothing spline (BSS) ANOVA framework. The BSS-ANOVA framework has several advantages over the traditional Gaussian process, including ease of handling categorical inputs and correlated outputs, and improved computational efficiency. Finally, this framework is then applied to the problem that motivated its design; a calibration of a computational fluid dynamics (CFD) model of a bubbling fluidized which is used as an absorber in a CO2capture system. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

This article develops a set of tools for smoothing and prediction with dependent point event patterns. The methodology is motivated by the problem of tracking weekly maps of violent crime events, but is designed to be straightforward to adapt to a wide variety of alternative settings. In particular, a Bayesian semiparametric framework is introduced for modeling correlated time series of marked spatial Poisson processes. The likelihood is factored into two independent components: the set of total integrated intensities and a series of process densities. For the former it is assumed that Poisson intensities are realizations from a dynamic linear model. In the latter case, a novel class of dependent stick-breaking mixture models are proposed to allow nonparametric density estimates to evolve in discrete time. This, a simple and flexible new model for dependent random distributions, is based on autoregressive time series of marginally beta random variables applied as correlated stick-breaking proportions. The approach allows for marginal Dirichlet process priors at each time and adds only a single new correlation term to the static model specification. Sequential Monte Carlo algorithms are described for online inference with each model component, and marginal likelihood calculations form the basis for inference about parameters governing temporal dynamics. Simulated examples are provided to illustrate the methodology, and we close with results for the motivating application of tracking violent crime in Cincinnati. [ABSTRACT FROM AUTHOR]

In this article, quantile regression methods are suggested for a class of smooth coefficient time series models. We use both local polynomial and local constant fitting schemes to estimate the smooth coefficients in a quantile framework. We establish the asymptotic properties of both the local polynomial and local constant estimators for α-mixing time series. Also, a bandwidth selector based on the nonparametric version of the Akaike information criterion is suggested, together with a consistent estimate of the asymptotic covariance matrix. Furthermore, the asymptotic behaviors of the estimators at boundaries are examined. A comparison of the local polynomial quantile estimator with the local constant estimator is presented. A simulation study is carried out to illustrate the performance of estimates. An empirical application of the model to real data further demonstrates the potential of the proposed modeling procedures. [ABSTRACT FROM AUTHOR]

In many practical problems, time series are realizations of nonstationary random processes. These processes can often be modeled as processes with slowly changing dynamics or as piecewise stationary processes. In these cases, various approaches to estimating the time-varying spectral density have been proposed. Our approach in this article is to estimate the log of the Dahlhaus local spectrum using a Bayesian mixture of splines. The basic idea of our approach is to first partition the data into small sections. We then assume that the log spectral density of the evolutionary process in any given partition is a mixture of individual log spectra. We use a mixture of smoothing splines model with time varying mixing weights to estimate the evolutionary log spectrum. The mixture model is fit using Markov chain Monte Carlo techniques that yield estimates of the log spectra of the individual subsections. In addition to an estimate of the local log spectral density, the method yields pointwise credible intervals. We use a reversible jump step to automatically determine the number of different spectral components. [ABSTRACT FROM AUTHOR]

This article focuses on variable selection for partially linear models when the covariates are measured with additive errors. We propose two classes of variable selection procedures, penalized least squares and penalized quantile regression, using the nonconvex penalized principle. The first procedure corrects the bias in the loss function caused by the measurement error by applying the so-called correction-for-attenuation approach, whereas the second procedure corrects the bias by using orthogonal regression. The sampling properties for the two procedures are investigated. The rate of convergence and the asymptotic normality of the resulting estimates are established. We further demonstrate that, with proper choices of the penalty functions and the regularization parameter, the resulting estimates perform asymptotically as well as an oracle property. Choice of smoothing parameters is also discussed. Finite sample performance of the proposed variable selection procedures is assessed by Monte Carlo simulation studies. We further illustrate the proposed procedures by an application. [ABSTRACT FROM AUTHOR]

Order selection is a fundamental and challenging problem in the application of finite mixture models. In this article, we develop a new penalized likelihood approach. The new method, modified smoothly clipped absolute deviation (MSCAD), deviates from information-based methods such as Akaike information criterion (AIC) and Bayesian information criterion (BIC) by introducing two penalty functions that depend on the mixing proportions and the component parameters. It is consistent at estimating both the order of the mixture model and the mixing distribution. Simulations show that MSCAD has much better performance than a number of existing methods. Two real-data examples are examined to illustrate the performance of MSCAD. [ABSTRACT FROM AUTHOR]

The smoothly clipped absolute deviation (SCAD) estimator, proposed by Fan and Li, has many desirable properties, including continuity, sparsity, and unbiasedness. The SCAD estimator also has the (asymptotically) oracle property when the dimension of covariates is fixed or diverges more slowly than the sample size. In this article we study the SCAD estimator in high-dimensional settings where the dimension of covariates can be much larger than the sample size. First we develop an efficient optimization algorithm that is fast and always converges to a local minimum. Second, we prove that the SCAD estimator still has the oracle property on high-dimensional problems. We perform numerical studies to compare the SCAD estimator with the LASSO and SIS-SCAD estimators in terms of prediction accuracy and variable selectivity when the true model is sparse. Through the simulation, we show that the variance, estimator of Fan and Li still works well for some limited high-dimensional cases where the true nonzero coefficients are not too small and the sample size is moderately large. We apply the proposed algorithm to analyze a high-dimensional microarray data set. [ABSTRACT FROM AUTHOR]

A common nonparametric approach to estimate the intensity function of an inhomogeneous spatial point process is through kernel smoothing. When conducting the smoothing, one typically uses events only in a local set around the point of interest. But the resulting estimator often is inconsistent, because the number of events in a fixed set is of order 1 for spatial point processes. In this article we propose a new covariate-based kernel smoothing method to estimate the intensity function. Our method defines the distance between any two points as the difference between their associated covariate values. Consequently, we determine the kernel weight for a given event of the process as a function of its new distance to the point of interest. Under some suitable conditions on the covariates and the spatial point process, we prove that our new estimator is consistent for the true intensity. To handle the situation with high-dimensional covariates, we also extend sliced inverse regression, a useful dimension-reduction tool in standard regression analysis, to spatial point processes. Simulations and an application to a real data example are used to demonstrate the usefulness of the proposed method. [ABSTRACT FROM AUTHOR]

In this article we propose a penalized clustering method for large-scale data with multiple covariates through a functional data approach. In our proposed method, responses and covariates are linked together through nonparametric multivariate functions (fixed effects), which have great flexibility in modeling various function features, such as jump points, branching, and periodicity. Functional ANOVA is used to further decompose multivariate functions in a reproducing kernel Hilbert space and provide associated notions of main effect and interaction. Parsimonious random effects are used to capture various correlation structures. The mixed-effects models are nested under a general mixture model in which the heterogeneity of functional data is characterized. We propose a penalized Henderson's likelihood approach for model fitting and design a rejection-controlled EM algorithm for the estimation. Our method selects smoothing parameters through generalized cross-validation. Furthermore, Bayesian confidence intervals are used to measure the clustering uncertainty. Simulation studies and real-data examples are presented to investigate the empirical performance of the proposed method. Open-source code is available in the R package MFDA. [ABSTRACT FROM AUTHOR]

SIMEX methods are attractive for solving curve estimation problems in errors-in-variables regression, using parametric or semiparametric techniques. However, nonparametric approaches are generally of quite a different type, being based on, for example, kernels, local-linear modeling, ridging, orthogonal series, or splines. All of these techniques involve the challenging (and not well studied) issue of empirical smoothing parameter choice. We show that SIMEX can be used effectively for selecting smoothing parameters when applying nonparametric methods to errors-in-variable regression. In particular, we suggest an approach based on multiple error-inflated (or remeasured) data sets and extrapolation. [ABSTRACT FROM AUTHOR]

The authors acknowledge the people who gave their comments on their article "Bivariate Binomial Spatial Modeling of Loa Ioa Prevalence in Tropical Africa" published within the journal. They agree to some comments on the spatial sparsity of the data used in their study. They think that the suggestion of using a hierarchical model with village and region random effects instead of continuous spatial smoothing is a good point.

We present a state-of-the-art application of smoothing for dependent bivariate binomial spatial data to Loa loa prevalence mapping in West Africa. This application starts with the nonspatial calibration of survey instruments, continues with the spatial model building and assessment, and ends with robust, tested software intended for use by field workers for online prevalence map updating. From a statistical perspective, we address several important methodological issues: building spatial models that are sufficiently complex to capture the structure of the data but remain computationally usable, reducing the computational burden in the handling of very large covariate data sets, and devising methods for comparing spatial prediction methods for a given exceedance policy threshold. [ABSTRACT FROM AUTHOR]

We introduce directional regression (DR) as a method for dimension reduction. Like contour regression, DR is derived from empirical directions, but achieves higher accuracy and requires substantially less computation. DR naturally synthesizes the dimension reduction estimators based on conditional moments, such as sliced inverse regression and sliced average variance estimation, and in doing so combines the advantages of these methods. Under mild conditions, it provides exhaustive and ,√n-consistent estimate of the dimension reduction space. We develop the asymptotic distribution of the DR estimator, and from that a sequential test procedure to determine the dimension of the central space. We compare the performance of DR with that of existing methods by simulation and find strong evidence of its advantage over a wide range of models. Finally, we apply DR to analyze a data set concerning the identification of hand-written digits. [ABSTRACT FROM AUTHOR]

In many fields of science, predicting variables of interest over a study region based on noisy data observed at some locations is an important problem. Two popular methods for the problem are kriging and smoothing splines. The former assumes that the underlying process is stochastic, whereas the latter assumes it is purely deterministic. Kriging performs better than smoothing splines in some situations, but is outperformed by smoothing splines in others. However, little is known regarding selecting between kriging and smoothing splines. In addition, how to perform variable selection in a geostatistical model has not been well studied. In this article we propose a general methodology for selecting among arbitrary spatial prediction methods based on (approximately) unbiased estimation of mean squared prediction errors using a data perturbation technique. The proposed method accounts for estimation uncertainty in both kriging and smoothing spline predictors, and is shown to be optimal in terms of two mean squared prediction error criteria. A simulation experiment is performed to demonstrate the effectiveness of the proposed methodology. The proposed method is also applied to a water acidity data set by selecting important variables responsible for water acidity based on a spatial regression model. Moreover, a new method is proposed for estimating the noise variance that is robust and performs better than some well-known methods. [ABSTRACT FROM AUTHOR]

We propose Bayesian parametric and semiparametric partially linear regression methods to analyze the outcome-dependent follow-up data when the random time of a follow-up measurement of an individual depends on the history of both observed longitudinal Outcomes and previous measurement times. We begin with the investigation of the simplifying assumptions of Lipsitz, Fitzmaurice, Ibrahim, Gelber, and Lipshultz, and present a new model for analyzing such data by allowing subject-specific correlations for the longitudinal response and by introducing a subject-specific latent variable to accommodate the association between the longitudinal measurements and the follow-up times. An extensive simulation study shows that our Bayesian partially linear regression method facilitates accurate estimation of the true regression line and the regression parameters. We illustrate our new methodology using data from a longitudinal observational study. [ABSTRACT FROM AUTHOR]

Surface estimation is important in many applications. When conventional smoothing procedures (e.g., running averages, local polynomial kernel smoothing procedures, smoothing spline procedures) are used for estimating jump surfaces from noisy data, jumps are blurred at the same time when noise is removed. In recent years, new smoothing methodologies have been proposed in the statistical literature for detecting jumps in surfaces and for estimating jump surfaces with jumps preserved. We provide a review of these methodologies. Because a monochrome image can be considered a jump surface of the image intensity function, with jumps at the outlines of objects, edge detection and image restoration problems in image processing are closely related to the jump surface estimation problem in statistics. We also review major methodologies on edge detection and image restoration, and discuss connections and differences among these methods and related methods in the statistical literature. [ABSTRACT FROM AUTHOR]

Time- and state-domain methods are two common approaches to nonparametric prediction. Whereas the former uses data predominantly from recent history, the latter relies mainly on historical information. Combining these two pieces of valuable information is an interesting challenge in statistics. We surmount this problem by dynamically integrating information from both the time and state domains. The estimators from these two domains are optimally combined based on a data-driven weighting strategy, which provides a more efficient estimator of volatility. Asymptotic normality is separately established for the time domain, the state domain, and the integrated estimators. By comparing the efficiency of the estimators, we demonstrate that the proposed integrated estimator uniformly dominates the other two estimators. The proposed dynamic integration approach is also applicable to other estimation problems in time series. Extensive simulations are conducted to demonstrate that the newly proposed procedure outperforms some popular ones, such as the RiskMetrics and historical simulation approaches, among others. In addition, empirical studies convincingly endorse our integration method. [ABSTRACT FROM AUTHOR]

A robust approach for testing the parametric form of a regression function versus an omnibus alternative is introduced. This generalizes existing robust methods for testing subhypotheses in a regression model. The new test is motivated by developments in modern smoothing-based testing procedures and can be viewed as a robustification of a smoothing-based conditional moment test. It is asymptotically normal under both the null hypothesis and local alternatives. The robustified test retains the "omnibus" property of the corresponding smoothing test; that is, it is consistent for any fixed smooth alternative in an infinite-dimensional space. It is shown that the bias of the asymptotic level under shrinking local contamination is bounded only if the second-order Hampel's influence function is bounded. The test's performance is demonstrated through both Monte Carlo simulations and application to an agricultural dataset. [ABSTRACT FROM AUTHOR]

In this article we consider the linear quantile regression model with a power transformation on the dependent variable. Like the classical Box-Cox transformation approach, it extends the applicability of linear models without resorting to nonparametric smoothing, but transformations on the quantile models are more natural due to the equivariance property of the quantiles under monotone transformations. We propose an estimation procedure and establish its consistency and asymptotic normality under some regularity conditions. The objective function employed in the estimation can also be used to check inadequacy of a power-transformed linear quantile regression model and to obtain inference on the transformation parameter. The proposed approach is shown to be valuable through illustrative examples. [ABSTRACT FROM AUTHOR]

Attachment loss, the extent of a tooth's root (in millimeters) that is no longer attached to surrounding bone by periodontal ligament, is often used to measure the current state of a patient's periodontal disease and monitor disease progression. Attachment loss data can be analyzed using a conditionally autoregressive (CAR) prior distribution, which smooths fitted values toward neighboring values. However, it may be desirable to have more than one class of neighbor relation in the spatial structure, so the different classes of neighbor relations can induce different degrees of smoothing. For example, we may wish to allow smoothing of neighbor pairs bridging the gap between teeth to be different from smoothing of pairs that do not bridge such gaps. Adequately modelling the spatial structure may improve monitoring of periodontal disease progression. This paper develops a two-neighbor-relation CAR model to handle this situation and presents associated theory to help explain the sometimes unusual posterior distributions of the parameters controlling the different types of smoothing. The posterior of these smoothing parameters often has long up- per tails, and its shape can change dramatically depending on the spatial structure. Like previous authors, we show that the prior distribution on these parameters has little effect on the posterior of the fixed effects, but has a marked influence on the posterior of both the random effects and the smoothing parameters. Our analysis of attachment loss data also suggests that the spatial structure itself varies between individuals. [ABSTRACT FROM AUTHOR]

We consider the problem of prediction for stationary and nonstationary univariate time series using a modification suggested by the usual exponentially weighted moving average method. The modification leads to a class of general exponential predictors that can improve on the usual finite approximations to an infinite autoregressive process. We provide the theoretical justifications and suggest a class of predictors that covers modified and finite autoregressive fits as special cases. Two examples involving sample data show how the method is competitive with autoregressive integrated moving average (ARIMA) when applied to a U.S. energy use series and improves on ARIMA when applied to a global temperature series. A simulation indicates that considerable improvements are possible for infinite autoregressive (ARIMA) processes exhibiting certain special patterns of long-range dependence. [ABSTRACT FROM AUTHOR]

Boston Harbor has a history of poor water quality, including contamination by enteric pathogens. We conduct a statistical analysis of data collected by the Massachusetts Water Resources Authority (MWRA) between 1996 and 2002 to evaluate the effects of court-mandated improvements in sewage treatment. Motivated by the ineffectiveness of standard Poisson mixture models and their zero-inflated counterparts, we propose a new negative binomial model for time series of Enterococcus counts in Boston Harbor, where nonstationarity and autocorrelation are modeled using a nonparametric smooth function of time in the predictor. Without further restrictions, this function is not identifiable in the presence of time-dependent covariates; consequently, we use a basis orthogonal to the space spanned by the covariates and use penalized quasi-likelihood (PQL) for estimation. We conclude that Enterococcus counts were greatly reduced near the Nut Island Treatment Plant (NITP) outfalls following the transfer of wastewaters from NITP to the Deer Island Treatment Plant (DITP) and that the transfer of wastewaters from Boston Harbor to the offshore diffusers in Massachusetts Bay reduced the Enterococcus counts near the DITP outfalls. [ABSTRACT FROM AUTHOR]

An easy-to-implement global procedure for testing the four assumptions of the linear model is proposed. The test can be viewed as a Neyman smooth test and relies only on the standardized residual vector. If the global procedure indicates a violation of at least one of the assumptions, then the components of the global test statistic can be used to gain insight into which assumptions have been violated. The procedure can also be used in conjunction with associated deletion statistics to detect unusual observations. Simulation results are presented indicating the sensitivity of the procedure in detecting model violations under a variety of situations, and its performance is compared with three potential competitors, including a procedure based on the Box-Cox power transformation. The procedure is demonstrated by applying it to a new car mileage dataset and a water salinity dataset that has been used earlier to illustrate model diagnostics. [ABSTRACT FROM AUTHOR]

We examine the effect of presmoothing functional data on estimating the dissimilarities among objects in a dataset, with applications to cluster analysis and other distance methods, such as multidimensional scaling and statistical matching. We prove that a shrinkage method of smoothing results in a better estimator of the dissimilarities among a set of noisy curves. For a model with independent noise structure, the smoothed-data dissimilarity estimator dominates the observed-data estimator. For a dependent-error model-often applicable when the functional data are measured nearly continuously over some domain-an asymptotic domination result is given for the smoothed-data estimator. A simulation study indicates the magnitude of improvement provided by the shrinkage estimator and examines its behavior for heavy-tailed noise structure. The shrinkage estimator presented here combines Stein estimation and basis function-based linear smoothers in a novel manner. Statisticians increasingly analyze sizable sets of functional data, and the results in this article are a useful contribution to the theory of the effect of presmoothing on functional data analysis. [ABSTRACT FROM AUTHOR]

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 co- variance 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. A key step is the derivation of asymptotic consistency and distribution results under mild conditions, using tools from functional analysis. Functional data analysis for sparse longitudinal data enables prediction of individual smooth trajectories even if only one or few measurements are available for a subject. Asymptotic pointwise and simultaneous confidence bands are obtained for predicted individual trajectories, based on asymptotic distributions, for simultaneous bands under the assumption of a finite number of components. Model selection techniques, such as the Akaike information criterion, are used to choose the model dimension corresponding to the number of eigenfunctions in the model. The methods are illustrated with a simulation study, longitudinal CD4 data for a sample of AIDS patients, and time-course gene expression data for the yeast cell cycle. [ABSTRACT FROM AUTHOR]

For the common binary response model, we propose a direct method for the nonparametric estimation of the effective dose level. The estimator is obtained by the composition of a nonparametric estimate of the quantile response curve and a classical density estimate. The new method yields a simple and reliable monotone estimate of the effective dose-level curve α ↠ EDα and is appealing to users of conventional smoothing methods as kernel estimators, local polynomials, series estimators, or smoothing splines. Moreover, it is computationally very efficient, because it does not require a numerical inversion of a monotonized estimate of the quantile dose-response curve. We prove asymptotic normality of the new estimate and compare it with an available alternative estimate (based on a monotonized nonparametric estimate of the dose-response curve and calculation of the inverse function) by means of a simulation study. [ABSTRACT FROM AUTHOR]

We consider multivariate temporal processes that are continuously observed within overlapping time windows. The intended application is in censored multistate and multivariate survival settings, where point processes are continuously observed. These data differ from other functional data, like longitudinal data, which are discretely observed at irregular times. Existing functional approaches to survival processes use intensity models, which require smoothing and depend critically on the choice of smoothing parameters, similarly to discretely observed data. In this article we study functional mean and association regression models for the point processes, with unspecified time-varying coefficients. The continuous observation scheme is exploited; the coefficients may be estimated nonparametrically by extending generalized estimating equations to continuously observed data. The estimators automatically converge at the parametric rate without smoothing, unlike with discretely observed data. Uniform consistency and weak convergence is established with empirical process techniques. The nonparametric estimators yield new tests for covariate effects, parametric submodeling of these effects, and goodness-of-fit testing. Simulation studies and an analysis of familial aggregation of alcoholism illustrate the methodology's practical utility. [ABSTRACT FROM AUTHOR]

We develop methods for performing smoothing computations in general state-space models. The methods rely on a particle representation of the filtering distributions, and their evolution through time using sequential importance sampling and resampling ideas. In particular, novel techniques are presented for generation of sample realizations of historical state sequences. This is carried out in a forward-filtering backward-smoothing procedure that can be viewed as the nonlinear, non-Gaussian counterpart of standard Kalman filter-based simulation smoothers in the linear Gaussian case. Convergence in the mean squared error sense of the smoothed trajectories is proved, showing the validity of our proposed method. The methods are tested in a substantial application for the processing of speech signals represented by a time-varying autoregression and parameterized in terms of time-varying partial correlation coefficients, comparing the results of our algorithm with those from a simple smoother based on the filtered trajectories. [ABSTRACT FROM AUTHOR]

In this article we propose a smoothing spline ANOVA model (SS-ANOVA) to estimate and to make inference on the time-varying log-spectrum of a locally stationary process. The time-varying spectrum is assumed to be smooth in both time and frequency. This assumption essentially turns a time-frequency spectral estimation problem into a 2-dimensional surface estimation problem. A smooth localized complex exponential (SLEX) basis is used to calculate the initial periodograms, and a SS-ANOVA is fitted to the log-periodograms. This approach allows the time and frequency domains to be modeled in a unified approach and jointly estimated. Inference procedures, such as confidence intervals, and hypothesis tests proposed for the SS-ANOVA can be adopted for the time-varying spectrum. Because of the smoothness assumption of the underlying spectrum, once we have the estimates on a time-frequency grid, we can calculate the estimate at any given time and frequency. This leads to a high computational efficiency, because for large datasets we need only estimate the initial raw periodograms at a much coarser grid. We study a penalized least squares estimator and a penalized Whittle likelihood estimator. The penalized Whittle likelihood estimator has smaller mean squared errors, whereas inference based on the penalized least squares method can adopt existing results. We present simulation results and apply our method to electroencephalogram data recorded during an epileptic seizure. [ABSTRACT FROM AUTHOR]

Recent results on discretization effects in unfolding intensity function of an indirectly observed nonhomogeneous Poisson process show that a critical feasibility condition for the strong L² consistency of the histogram sieve maximum likelihood estimates requires that the approximation error of the function of interest with step functions tend to 0 faster than the squared singular values of the discretized folding operator. This condition may not be fulfilled in some standard inverse problems like, for example, the deconvolution problem and the Wicksell problem discretized in a standard way. The condition may be satisfied, however, with suitably modified discrete operator and suitably constructed parametric sets for the discrete problems. Motivated by these results, an additional smoothing step is added to the EMS (expectation-maximization-smoothing) algorithm and an automatic procedure for the choice of a smoothing parameter is proposed. An application to the Wicksell problem is presented. In this context, the theory also suggests a special, nonuniform binning in the data space. A simulation study demonstrates that the new approach considerably reduces the L² risk when compared to both the standard EMS algorithm and to a spectral procedure specially designed for the Wicksell problem. A priori knowledge of the value of the estimated function at 0 may further reduce the risk very significantly. Theoretical results on the strong L² consistency and convergence rates are given. [ABSTRACT FROM AUTHOR]

We investigate the estimation of the conditional quantile when many covariates are involved. In particular, we model the conditional quantile of a response as a nonlinear additive function of relevant covariates. For this setup, we propose a nonparametric smoother to estimate the unknown functions. The estimator provides direct computation of the nonlinear functions. Because it does not require any iteration, the estimator allows fast and routine data analysis. On the theoretical front, we also show asymptotic properties of the estimator, including mean squared error and limiting distribution. The theory confirms that for moderate dimension of the covariates, the estimator escapes the "curse of dimensionality" problem. Both simulated and real data examples are provided to illustrate the methodology. [ABSTRACT FROM AUTHOR]

Situations with incomplete multivariate spatial data on a lattice are considered. The goal is to impute the missing data in the presence of edges or boundaries and recover the image. Two methods based on Bayesian hierarchical models that iterate between edge detection and spatial smoothing to impute the missing data within identified homogeneous regions are examined. Their performance is compared with another method that imputes the missing values using edge-preserving spatial smoothers with locally varying weights. The performances of the three methods are compared on artificial and real datasets. It is seen that information from the multivariate data is critical in recovering the images. An application with color images where only one of three primary colors (red, green, or blue) is observed at each pixel is used to illustrate the results. [ABSTRACT FROM AUTHOR]

Nonparametric regression, often called smoothing, is a widely used data analysis method. The use of a smoother requires the choice of a smoothing parameter that by balancing fidelity and roughness controls how much smoothing is done. Two popular selection criteria for choosing the smooching parameter are Cp and generalized maximum likelihood (GML). Each of these has its own problems. For Cp, the problem is its high variability, whereas for GML, the problem is its potentially large bias. By studying the geometry of selection criteria, we give an intuitive explanation of the strength and weakness of Cp and CML. The geometry then motivates a new selection method, the extended exponential (EE) criterion, which combines the strength of Cp and GMI, but mitigates their weaknesses in terms of variability, bias, and undersmoothing. [ABSTRACT FROM AUTHOR]

Longitudinal samples, i.e., datasets with repeated measurements over time, are common in biomedical and epidemiological studies such as clinical trials and cohort observational studies. An exploratory tool for the analyses of such data is the varying coefficient model Y(t) =XT (t)β(t) + ϵ(t), where Y(t) and X(t) = (X(0) (t). . . . .X(k)(t))T are the response and covariates at time t, β(t) = (β0(t). . . . .βk(t))T are smooth coefficient curves of t and ϵ(t) is a mean zero stochastic process. A special case that is of particular interest in many situations is data with time-dependent response and time-independent covariates. We propose in this article a component wise smoothing spline method for estimating β0(t). . . . .βk(t) nonparametrically based on the previous varying coefficient model and a longitudinal sample of (t, Y (t), X) with time-independent covariates X = (X(0). . . . .X(k))T from n independent subjects. A "leave-one-subject-out" cross-validation is suggested to choose the smoothing parameters. Asymptotic properties of our spline estimators are developed through the explicit expressions of their asymptotic normality and risk representations, which provide useful insights tor inferences. Applications and finite sample properties of our procedures are demonstrated through a longitudinal sample of opioid detoxification and a simulation study. [ABSTRACT FROM AUTHOR]

Numerous nonparametric regression methods exist that yield consistent estimators of function curves. Often, one is also interested in constructing confidence intervals for the unknown function. When a function estimate is based on a single global smoothing parameter the resulting confidence intervals may hold their desired confidence level 1 - α on average but because bias in nonparametric estimation is not uniform, they do not hold the desired level uniformly at all design points. Most research in this area has focused on mean squared error properties of the estimator, for example MISE, itself a global measure. In addition, measures like MISE are one step removed from the practical issue of coverage probability. Recent work that focuses on coverage probability has considered only coverage in an average sense, ignoring the important issue of uniformity of coverage across the design space. To deal with this problem, a new estimator is developed which uses a local cross-validation criterion (LVC) to determine a separate smoothing parameter for each design point. The local smoothing parameters are then used to compute the point estimators of the regression curve and the corresponding pointwise confidence intervals. Incorporation of local information through the new method is shown, via Monte Carlo simulation, to yield more uniformly valid pointwise confidence intervals for nonparametric regression curves. Diagnostic plots are developed (Breakout plots) to visually inspect the degree of uniformity of coverage of the confidence intervals. The approach, here applied to cubic smoothing splines, easily generalizes to many other nonparametric regression estimators. The improved curve estimation is not a solely theoretical improvement such as providing an estimators. The improved curve estimation is not a solely theoretical improvement such s providing an estimator that has a faster EASE convergence rate but shows its worth empirically by yielding improved coverage probabilities through reliable pointwise confidence intervals. [ABSTRACT FROM AUTHOR]

We combine a smoothing spline analysis of variance (SS-ANOVA) model and a log-linear model to build a partly flexible model for multivariate Bernoulli data. The joint distribution conditioning on the predictor variables is estimated. The log odds ration is used to measure the association between outcome variables. A numerical scheme based on the block one-step successive over relaxation SOR-Newton-Ralphson algorithm is proposed to obtain an approximate solution for the variational problem. We extend the generalized approximate cross validation (GACV) and the randomized GACV for choosing smoothing parameters to the case of multivariate Bernoulli responses. The randomized version is fast and stable to compute and is used to adaptively select smoothing parameters in each block one-step SOR iteration. Approximate Bayesian confidence intervals are obtained for the flexible estimates of the conditional logit functions. Simulation studies are conducted to check the performance of the proposed method, using the comparative Kullback-Leibler distance as a yardstick. Finally, the model is applied to two-eye observational date from the Beaver Dam Study, to examine the association of pigmentary abnormalities and various coveriates. [ABSTRACT FROM AUTHOR]

In the use of smoothing methods in data analysis, an important question is which observed features are "really there," as opposed to being spurious sampling artifacts. An approach is described based on scale-space ideas originally developed in the computer vision literature. Assessment of SIgnificant ZERo crossings of derivatives results in the SiZer map, a graphical device for display of significance of features with respect to both location and scale. Here "scale" means "level of resolution"; that is, "bandwidth." [ABSTRACT FROM AUTHOR]

Presents the authors' response on the comments made by various author regarding their article entitled `Edge-Preserving Smoothers for Image Processing. Response to Douglas Simpson, Xuming He and Yao Tsorng Liu; Agreement with the main point that image understanding is different from image denoising; Discussion of an example where image denoising and M smoothing approach are useful.

Opinion. Comments on the article written by C.K. Chu, I.K. Glad, F. Godtliebsen and J.S. Marron regarding edge-preserving smoothers for image processing. History of M estimation in the statistical literature; Difference of the article from previous works; Presentation of a preliminary results using a penalty function.

Develops prior distributions for histogram inference favoring smooth population frequencies that is probability vectors with small differences for neighboring categories. Formalization of the concept of linearly filtering a random vector to obtain a `filtered variate' generalization; Development of the filtered-variate Dirichlet family of prior distribution.

Proposes a monotone B-spline smoothing based on L1 optimization built on the ideas of Koenker, Ng and Portnoy and Ramsay. Description of monotone B-spline and knot selection; Indicative asymptotic rates of convergence results; Invstigation of the finite-sample performance of the monotone B-spline smoother.

Introduces a method of selecting the smoothing parameters of nonparametric regression estimators termed one-sided cross-validation (OSCV). List of methods that can be used in nonparametric estimation Description of the method and its use on a real dataset; Comparison of OSCV with existing method; Asymptotic variances of one-sided and ordinary cross-validation.