Unified Inference for Sparse and Dense Longitudinal Data in Time‐varying Coefficient Models

Time‐varying coefficient models are widely used in longitudinal data analysis. These models allow the effects of predictors on response to vary over time. In this article, we consider a mixed‐effects time‐varying coefficient model to account for the within subject correlation for longitudinal data. We show that when kernel smoothing is used to estimate the smooth functions in time‐varying coefficient models for sparse or dense longitudinal data, the asymptotic results of these two situations are essentially different. Therefore, a subjective choice between the sparse and dense cases might lead to erroneous conclusions for statistical inference. In order to solve this problem, we establish a unified self‐normalized central limit theorem, based on which a unified inference is proposed without deciding whether the data are sparse or dense. The effectiveness of the proposed unified inference is demonstrated through a simulation study and an analysis of Baltimore MACS data.     

Transformations for Smooth Regression Models with Multiplicative Errors

We consider whether one should transform to estimate nonparametrically a regression curve sampled from data with a constant coefficient of variation, i.e. with multiplicative errors. Kernel-based smoothing methods are used to provide curve estimates from the data both in the original units and after transformation. Comparisons are based on the mean-squared error (MSE) or mean integrated squared error (MISE), calculated in the original units. Even when the data are generated by the simplest multiplicative error model, the asymptotically optimal MSE (or MISE) is surprisingly not always obtained by smoothing transformed data, but in many cases by directly smoothing the original data. Which method is optimal depends on both the regression curve and the distribution of the errors. Data-based procedures which could be useful in choosing between transforming and not transforming a particular data set are discussed. The results are illustrated on simulated and real data.     

An exploratory data analysis in scale-space for interval-valued data

We propose an exploratory data analysis approach when data are observed as intervals in a nonparametric regression setting. The interval-valued data contain richer information than single-valued data in the sense that they provide both center and range information of the underlying structure. Conventionally, these two attributes have been studied separately as traditional tools can be readily used for single-valued data analysis. We propose a unified data analysis tool that attempts to capture the relationship between response and covariate by simultaneously accounting for variability present in the data. It utilizes a kernel smoothing approach, which is conducted in scale-space so that it considers a wide range of smoothing parameters rather than selecting an optimal value. It also visually summarizes the significance of trends in the data as a color map across multiple locations and scales. We demonstrate its effectiveness as an exploratory data analysis tool for interval-valued data using simulated and real examples.     

Discrete regularized discriminant analysis

A method of regularized discriminant analysis for discrete data, denoted DRDA, is proposed. This method is related to the regularized discriminant analysis conceived by Friedman (1989) in a Gaussian framework for continuous data. Here, we are concerned with discrete data and consider the classification problem using the multionomial distribution. DRDA has been conceived in the small-sample, high-dimensional setting. This method has a median position between multinomial discrimination, the first-order independence model and kernel discrimination. DRDA is characterized by two parameters, the values of which are calculated by minimizing a sample-based estimate of future misclassification risk by cross-validation. The first parameter is acomplexity parameter which provides class-conditional probabilities as a convex combination of those derived from the full multinomial model and the first-order independence model. The second parameter is asmoothing parameter associated with the discrete kernel of Aitchison and Aitken (1976). The optimal complexity parameter is calculated first, then, holding this parameter fixed, the optimal smoothing parameter is determined. A modified approach, in which the smoothing parameter is chosen first, is discussed. The efficiency of the method is examined with other classical methods through application to data.     

Global optimization of the generalized cross-validation criterion

Generalized cross-validation is a method for choosing the smoothing parameter in smoothing splines and related regularization problems. This method requires the global minimization of the generalized cross-validation function. In this paper an algorithm based on interval analysis is presented to find the globally optimal value for the smoothing parameter, and a numerical example illustrates the performance of the algorithm.     

Relative smoothing of discrete distributions with sparse observations

Quite often we are faced with a sparse number of observations over a finite number of cells and are interested in estimating the cell probabilities. Some local polynomial smoothers or local likelihood estimators have been proposed to improve on the histogram, which would produce too many zero values. We propose a relativized local polynomial smoothing for this problem, weighting heavier the estimating errors in small probability cells. A simulation study about the estimators that are proposed show a good behaviour with respect to natural error criteria, especially when dealing with sparse observations.     

Semiparametric Time-Varying Coefficients Regression Model for Longitudinal Data

Abstract.  In this paper, we consider a semiparametric time-varying coefficients regression model where the influences of some covariates vary non-parametrically with time while the effects of the remaining covariates follow certain parametric functions of time. The weighted least squares type estimators for the unknown parameters of the parametric coefficient functions as well as the estimators for the non-parametric coefficient functions are developed. We show that the kernel smoothing that avoids modelling of the sampling times is asymptotically more efficient than a single nearest neighbour smoothing that depends on the estimation of the sampling model. The asymptotic optimal bandwidth is also derived. A hypothesis testing procedure is proposed to test whether some covariate effects follow certain parametric forms. Simulation studies are conducted to compare the finite sample performances of the kernel neighbourhood smoothing and the single nearest neighbour smoothing and to check the empirical sizes and powers of the proposed testing procedures. An application to a data set from an AIDS clinical trial study is provided for illustration.     

Robust optimal estimation of location from discretely sampled functional data

Estimating location is a central problem in functional data analysis, yet most current estimation procedures either unrealistically assume completely observed trajectories or lack robustness with respect to the many kinds of anomalies one can encounter in the functional setting. To remedy these deficiencies we introduce the first class of optimal robust location estimators based on discretely sampled functional data. The proposed method is based on M-type smoothing spline estimation with repeated measurements and is suitable for both commonly and independently observed trajectories that are subject to measurement error. We show that under suitable assumptions the proposed family of estimators is minimax rate optimal both for commonly and independently observed trajectories and we illustrate its highly competitive performance and practical usefulness in a Monte-Carlo study and a real-data example involving recent Covid-19 data.     

Smoothing Splines and Shape Restrictions

Constrained smoothing splines are discussed under order restrictions on the shape of the function m . We consider shape constraints of the type m ^{( r )}≥ 0, i.e. positivity, monotonicity, convexity, .... (Here for an integer r ≥ 0, m ^{( r )} denotes the r th derivative of m .) The paper contains three results: (1) constrained smoothing splines achieve optimal rates in shape restricted Sobolev classes; (2) they are equivalent to two step procedures of the following type: (a) in a first step the unconstrained smoothing spline is calculated; (b) in a second step the unconstrained smoothing spline is "projected" onto the constrained set. The projection is calculated with respect to a Sobolev-type norm; this result can be used for two purposes, it may motivate new algorithmic approaches and it helps to understand the form of the estimator and its asymptotic properties; (3) the infinite number of constraints can be replaced by a finite number with only a small loss of accuracy, this is discussed for estimation of a convex function.     

Kernel Smoothing Density Estimation when Group Membership is Subject to Missing

Density function is a fundamental concept in data analysis. Non-parametric methods including kernel smoothing estimate are available if the data is completely observed. However, in studies such as diagnostic studies following a two-stage design the membership of some of the subjects may be missing. Simply ignoring those subjects with unknown membership is valid only in the MCAR situation. In this paper, we consider kernel smoothing estimate of the density functions, using the inverse probability approaches to address the missing values. We illustrate the approaches with simulation studies and real study data in mental health.     

Optimal smoothing of direct estimates of the power spectrum

In this paper we demonstrate how the task of spectral density estimation by direct methods may be posed as that of solving a simple optimal smoothing problem. A criterion functional is considered which involves a smoothness frequency domain term and a fidelity time domain term.     

Automatic locally adaptive smoothing for tree-based set estimation

Tree-based methods similar to CART have recently been utilized for problems in which the main goal is to estimate some set of interest. It is often the case that the boundary of the true set is smooth in some sense, however tree-based estimates will not be smooth, as they will be a union of ‘boxes’. We propose a general methodology for smoothing such sets that allows for varying levels of smoothness on the boundary automatically. The method is similar to the idea underlying support vector machines, which is applying a computationally simple technique to data after a non-linear mapping to produce smooth estimates in the original space. In particular, we consider the problem of level-set estimation for regression functions and the dyadic tree-based method of Willett and Nowak [Minimax optimal level-set estimation, IEEE Trans. Image Process. 16 (2007), pp. 2965–2979].     

The use of Smooth Bootstrap Techniques for Estimating the Error Rate of a Prediction Rule

In this paper we present a simulation study for comparing differents methods for estimating the prediction error rate in a discrimination problem. We consider the Cross-validation, Bootstrap and Bayesian Bootstrap methods for such as problem, while also elaborating on both simple and Bayesian Bootstrap methods by smoothing techniques. We observe as the smoothing procedure lead to improvements in the estimation of the true error rate of the discrimination rule, specially in the case of the smooth Bayesian Bootstrap estimator, whose reduction in M.S.E. resulted from the high positive correlation between the true error rate and its estimations based in this method.     

Multiscale local polynomial decompositions using bandwidths as scales

The multiscale local polynomial transform, developped in this paper, combines the benefits from local polynomial smoothing with sparse multiscale decompositions. The contribution of the paper is twofold. First, it focusses on the bandwidths used throughout the transform. These bandwidths operate as user controlled scales in a multiscale analysis, which is explained to be of particular interest in the case of nonequispaced data. The paper presents both a likelihood based optimal bandwidth selection and a fast, heuristic approach. The second contribution of the paper is the combination of local polynomial smoothing with orthogonal prefilters, similar to Daubechies’ wavelet filters, but defined on irregularly spaced covariate values.     

Numerical experiments with the thin plate histospline

The thin plate volume matching and volume smoothing histosplines are described. These histosplines are suitable for estimating densities or incidence rates as a function of position on the plane when data is aggregated by area, for example by county. We give a numerical algorithm for the volume matching histospline and for the volume smoothing histospline using generalized cross validation to estimate the smoothing parameter. Some numerical experiments were performed using synthetic data, population data and SMR's (standardized mortality ratios) aggregated by county over the state of Wisconsin. The method turns out to be not particularly suited for obtaining population density maps where the population density can vary by two orders of magnitude, because the histospline can be negative in     

Automatic and asymptotically optimal data sharpening for nonparametric regression

In this article we consider data-sharpening methods for nonparametric regression. In particular modifications are made to existing methods in the following two directions. First, we introduce a new tuning parameter to control the extent to which the data are to be sharpened, so that the amount of sharpening is adaptive and can be tuned to best suit the data at hand. We call this new parameter the sharpening parameter. Second, we develop automatic methods for jointly choosing the value of this sharpening parameter as well as the values of other required smoothing parameters. These automatic parameter selection methods are shown to be asymptotically optimal in a well defined sense. Numerical experiments were also conducted to evaluate their finite-sample performances. To the best of our knowledge, there is no bandwidth selection method developed in the literature for sharpened nonparametric regression.     

ACCELERATED FAILURE TIME MODELS WITH NONLINEAR COVARIATES EFFECTS

As a flexible alternative to the Cox model, the accelerated failure time (AFT) model assumes that the event time of interest depends on the covariates through a regression function. The AFT model with non‐parametric covariate effects is investigated, when variable selection is desired along with estimation. Formulated in the framework of the smoothing spline analysis of variance model, the proposed method based on the Stute estimate ( Stute, 1993 [Consistent estimation under random censorship when covariables are present, J. Multivariate Anal. 45 , 89–103]) can achieve a sparse representation of the functional decomposition, by utilizing a reproducing kernel Hilbert norm penalty. Computational algorithms and theoretical properties of the proposed method are investigated. The finite sample size performance of the proposed approach is assessed via simulation studies. The primary biliary cirrhosis data is analyzed for demonstration.     

Global and local statistical properties of fixed-length nonparametric smoothers

The main purpose of this study is to analyze the global and local statistical properties of nonparametric smoothers subject to a priori fixed length restriction. In order to do so, we introduce a set of local statistical measures based on their weighting system shapes and weight values. In this way, the local statistical measures of bias, variance and mean square error are intrinsic to the smoothers and independent of the data to which they will be applied on. One major advantage of the statistical measures relative to the classical spectral ones is their easiness of calculation. However, in this paper we use both in a complementary manner. The smoothers studied are based on two broad classes of weighting generating functions, local polynomials and probability distributions. We consider within the first class, the locally weighted regression smoother (loess) of degree 1 and 2 (L1 and L2), the cubic smoothing spline (CSS), and the Henderson smoothing linear filter (H); and in the second class, the Gaussian kernel (GK). The weighting systems of these estimators depend on a smoothing parameter that traditionally, is estimated by means of data dependent optimization criteria. However, by imposing to all of them the condition of an equal number of weights, it will be shown that some of their optimal statistical properties are no longer valid. Without any loss of generality, the analysis is carried out for 13- and 9-term lengths because these are the most often selected for the Henderson filters in the context of monthly time series decomposition. We would like to thank an Associate Editor and an anonymous referee for their valuable comments on an earlier version of this paper. Financing from MURST is also gratefully acknowledged.     

A goodness-of-fit test for logistic-normal models using nonparametric smoothing method

Logistic-normal models can be applied for analysis of longitudinal binary data. The aim of this article is to propose a goodness-of-fit test using nonparametric smoothing techniques for checking the adequacy of logistic-normal models. Moreover, the leave-one-out cross-validation method for selecting the suitable bandwidth is developed. The quadratic form of the proposed test statistic based on smoothing residuals provides a global measure for checking the model with categorical and continuous covariates. The formulae of expectation and variance of the proposed statistics are derived, and their asymptotic distribution is approximated by a scaled chi-squared distribution. The power performance of the proposed test for detecting the interaction term or the squared term of continuous covariates is examined by simulation studies. A longitudinal dataset is utilized to illustrate the application of the proposed test.     

Penalized least squares smoothing of two-dimensional mortality tables with imposed smoothness

This paper presents a method to estimate mortality trends of two-dimensional mortality tables. Comparability of mortality trends for two or more of such tables is enhanced by applying penalized least squares and imposing a desired percentage of smoothness to be attained by the trends. The smoothing procedure is basically determined by the smoothing parameters that are related to the percentage of smoothness. To quantify smoothness, we employ an index defined first for the one-dimensional case and then generalized to the two-dimensional one. The proposed method is applied to data from member countries of the OECD. We establish as goal the smoothed mortality surface for one of those countries and compare it with some other mortality surfaces smoothed with the same percentage of two-dimensional smoothness. Our aim is to be able to see whether convergence exists in the mortality trends of the countries under study, in both year and age dimensions.