Local Linear Smoothing Research Articles

A novel robust approach for analysis of longitudinal data

A new robust estimating equation approach for analysis of longitudinal data is developed. To achieve robustness against outliers, a novel approach which corrects the bias induced by outliers through centralizing the covariate matrix in the estimating equation is proposed. The covariates are centralized by subtracting their conditional expectations and the conditional expectations can be estimated by using the local linear smoothing method. The consistency and asymptotic normality of the proposed estimator are established under some regularity conditions. Extensive simulation studies show that the proposed method is robust, has a high efficiency, and is not limited to some specific error distributions. In the end, the proposed method is applied to the longitudinal study of prevalent patients with type 2 diabetes and confirms the effectiveness of dietary fibre intake in reducing glycolated hemoglobin A1c level.

Analysis of longitudinal data with semiparametric varying-coefficient mean–covariance models

The efficient estimation of regression coefficients in the longitudinal data analysis requires a correct specification of the covariance structure. Existing approaches usually focus on modeling the mean with specification of certain covariance structures, which may lead to inefficient or biased estimators of parameters in the mean if misspecification occurs. In this article, we propose a novel data-driven approach based on semiparametric varying-coefficient models to model the mean and the covariance simultaneously, motivated by the modified Cholesky decomposition. An iterative estimation method is proposed, consisting of an orthogonality-based technique for parameters, an adaptive jump-preserving estimation method for varying coefficients, a modification of local linear smoothing technique for the autoregressive coefficient function, and a kernel smoothing technique for the variance function. Theoretical properties of the resulting estimators including uniform consistency and asymptotic normality are explicitly studied under certain mild conditions. Simulation studies are carried out to evaluate the efficacy of the proposed methods, and an analysis of a real data example is provided for illustration.

Sparse model identification and learning for ultra-high-dimensional additive partially linear models

The additive partially linear model (APLM) combines the flexibility of nonparametric regression with the parsimony of regression models, and has been widely used as a popular tool in multivariate nonparametric regression to alleviate the “curse of dimensionality”. A natural question raised in practice is the choice of structure in the nonparametric part, i.e., whether the continuous covariates enter into the model in linear or nonparametric form. In this paper, we present a comprehensive framework for simultaneous sparse model identification and learning for ultra-high-dimensional APLMs where both the linear and nonparametric components are possibly larger than the sample size. We propose a fast and efficient two-stage procedure. In the first stage, we decompose the nonparametric functions into a linear part and a nonlinear part. The nonlinear functions are approximated by constant spline bases, and a triple penalization procedure is proposed to select nonzero components using adaptive group LASSO. In the second stage, we refit data with selected covariates using higher order polynomial splines, and apply spline-backfitted local-linear smoothing to obtain asymptotic normality for the estimators. The procedure is shown to be consistent for model structure identification. It can identify zero, linear, and nonlinear components correctly and efficiently. Inference can be made on both linear coefficients and nonparametric functions. We conduct simulation studies to evaluate the performance of the method and apply the proposed method to a dataset on the Shoot Apical Meristem (SAM) of maize genotypes for illustration.

Pairwise directions estimation for multivariate response regression data

ABSTRACTThis article concerns the analysis of multivariate response data with multi-dimensional covariates. Based on local linear smoothing techniques, we propose an iteratively adaptive estimation method to reduce the dimensions of response variables and covariates. Two weighted estimation strategies are incorporated in our approach to provide initial estimates. Our proposal is also extended to curve response data for a data-adaptive basis function searching. Instead of focusing on goodness of fit, we shift the problem to reveal the data structure and basis patterns. Simulation studies with multivariate response and curve data are conducted for our pairwise directions estimation (PDE) approach in comparison with sliced inverse regression of Li et al. [Dimension reduction for multivariate response data. J Amer Statist Assoc. 2003;98:99–109]. The results demonstrate that the proposed PDE method is useful for data with responses approximating linear or bending structures. Illustrative applications to two real datasets are also presented.

Additive partially linear models for ultra‐high‐dimensional regression

We consider a semiparametric additive partially linear regression model (APLM) for analysing ultra‐high‐dimensional data where both the number of linear components and the number of non‐linear components can be much larger than the sample size. We propose a two‐step approach for estimation, selection, and simultaneous inference of the components in the APLM. In the first step, the non‐linear additive components are approximated using polynomial spline basis functions, and a doubly penalized procedure is proposed to select nonzero linear and non‐linear components based on adaptive lasso. In the second step, local linear smoothing is then applied to the data with the selected variables to obtain the asymptotic distribution of the estimators of the nonparametric functions of interest. The proposed method selects the correct model with probability approaching one under regularity conditions. The estimators of both the linear part and the non‐linear part are consistent and asymptotically normal, which enables us to construct confidence intervals and make inferences about the regression coefficients and the component functions. The performance of the method is evaluated by simulation studies. The proposed method is also applied to a dataset on the shoot apical meristem of maize genotypes.

Discrimination of dynamically and post‐dynamically recrystallized grains based on EBSD data: application to Inconel 718

The importance of angular resolution in EBSD analyses is discussed based on an Inconel 718 sample containing several populations of recrystallized grains, with subtle differences in dislocation contents. Classical EBSD analyses (with angular resolution in the range of 0.5-1°) do not allow for distinguishing recrystallized grains grown dynamically or post-dynamically. The angular resolution of EBSD orientation and misorientation data can be significantly improved (down to about 0.1-0.2°) either using more sophisticated Kikuchi pattern indexing methods and/or using the recently proposed LLASS denoising filter (Local Linear Automatic Smoothing Splines). Then the coexistence of both dynamically and post-dynamically recrystallized grains in the sample can be confirmed and quantified. ECCI images unambiguously confirm the conclusions drawn from the analysis of improved angular resolution EBSD data, and furthermore reveal the presence of thermal stress induced dislocations with typical patterns in water quenched Inconel 718 recrystallized grains. LAY DESCRIPTION: EBSD is widely used to study recrystallization phenomena. Conventional EBSD is nevertheless not able to distinguish dynamic recrystallized grains from post-dynamic recrystallized grains which differ by subtitle differences in dislocation contents. In this paper, we show that improving the orientation precision of EBSD data by means of different methods allows distinguishing these two recrystallized grains populations. Analyses and discussion are based on an Inconel 718, a famous Nickel-based superalloy in aeronautic.

A partial-robust-ridge-based regression model with fuzzy predictors-responses

This paper extends the conventional semi-parametric partial linear regression model with fuzzy predictors and fuzzy response in cases where 1—outliers occur in data set, 2—multicollinearity exists in predictors. For this purpose, the classical kernel fitting method is first extended for estimating unknown fuzzy smooth function. Then, inspired by the conventional local linear smoothing method, a kernel-based weighted criterion is suggested for determining the exact coefficients on the basis of ridge methodology. Some common goodness-of-fit criteria are also used to examine the performance of the proposed method. The effectiveness of the proposed method is then illustrated through two numerical examples including a simulation study. The proposed method is also compared with some common fuzzy multiple regression models with fuzzy predictors and fuzzy responses. The numerical results clearly indicate that 1—the proposed method is not sensitive to the outliers, 2—the proposed method provides sufficiently accurate results in cases where multicollinearity happens among predictors and 3—comparing to existing fuzzy multiple regressions, the proposed model provides more efficient predictions. • A fuzzy semi-parametric regression model is proposed when multicollinearity and/or outliers occur. • The fuzzy model contains the fuzzy predictors-responses. • The ridge methodology and a novel robust method are applied to evaluate crisp coefficients. • The method is compared with several methods using various goodness-of-fit-measures.

Two-step estimation of time-varying additive model for locally stationary time series

In the analysis of locally stationary process, a time-varying additive model (tvAM) can effectively capture the dynamic feature of regression function. In combination with the strengths of tensor product of B-spline smoothing and local linear smoothing method, a two-step estimation method is proposed. It is shown that the proposed estimator is uniformly consistent and asymptotically oracle efficient as if the other component functions were known. Furthermore, a nonparametric bootstrap procedure is proposed to test the time-varying property of regression function. Simulation studies investigate the finite-sample performance of the proposed methods and validate the asymptotic theory. An environmental dataset illustrating the proposed method is also considered.

Option Implied Risk-Neutral Density Estimation: A Robust and Flexible Method

In practice, a reliable and flexible estimation of risk-neutral density from empirical data is a challenging task since it can not be observed directly from the market. In this study, we apply Bernstein polynomial basis to recover the risk-neutral density function from the observed price quotes of European-type option contingent on an underlying asset. More importantly, we perform an extensive simulation study to examine the flexibility and robustness of the proposed method in recovering different shapes of the true risk-neutral density function from noisy option price quotes. Also, we compare the proposed method with other two popular nonparametric methods namely the constrained local linear polynomial smoothing and the smoothed implied volatility smile reported in the literature. Accuracy and stability of the three nonparametric methods are assessed by the root mean integrated square error criterion. The simulation results show that the proposed method is flexible as it exhibits the various shapes of the true risk-neutral density function even when the volatility is high. Moreover, in comparison with the other two methods, the proposed approach is robust and yields more accurate densities even in the presence of noise. Finally, we demonstrate the applicability of the proposed method in recovering a smooth risk-neutral density function from the S&P 500 market index option data.

SIMEX estimation for single-index model with covariate measurement error

In this paper, we consider the single-index measurement error model with mismeasured covariates in the nonparametric part. To solve the problem, we develop a simulation-extrapolation (SIMEX) algorithm based on the local linear smoother and the estimating equation. For the proposed SIMEX estimation, it is not needed to assume the distribution of the unobserved covariate. We transform the boundary of a unit ball in $\mathbb{R}^p$ to the interior of a unit ball in $\mathbb{R}^{p-1}$ by using the constraint $\|\beta\|=1$. The proposed SIMEX estimator of the index parameter is shown to be asymptotically normal under some regularity conditions. We also derive the asymptotic bias and variance of the estimator of the unknown link function. Finally, the performance of the proposed method is examined by simulation studies and is illustrated by a real data example.

A plug-in bandwidth selector for nonparametric quantile regression

In the framework of quantile regression, local linear smoothing techniques have been studied by several authors, particularly by Yu and Jones (J Am Stat Assoc 93:228–237, 1998). The problem of bandwidth selection was addressed in the literature by the usual approaches, such as cross-validation or plug-in methods. Most of the plug-in methods rely on restrictive assumptions on the quantile regression model in relation to the mean regression, or on parametric assumptions. Here we present a plug-in bandwidth selector for nonparametric quantile regression that is defined from a completely nonparametric approach. To this end, the curvature of the quantile regression function and the integrated squared sparsity (inverse of the conditional density) are both nonparametrically estimated. The new bandwidth selector is shown to work well in different simulated scenarios, particularly when the conditions commonly assumed in the literature are not satisfied. A real data application is also given.

Optimal weighting schemes for longitudinal and functional data

We propose optimal weighting schemes for both mean and covariance estimations for functional data based on local linear smoothing such that the L2 rate of convergence is minimized. These schemes can self-adjust to the sampling plan and lead to practical improvements.

Phase I monitoring with nonparametric mixed‐effect models

AbstractIn many real‐life applications, the quality of products from a process is monitored by a functional relationship between a response variable and one or more explanatory variables. In these applications, methodologies of profile monitoring are used to check the stability of this relationship over time. In phase I of profile monitoring, historical data points that can be represented by curves (or profiles) are collected. In this article, 2 procedures are proposed for detecting outlying profiles in phase I data, by incorporating the local linear kernel smoothing within the framework of nonparametric mixed‐effect models. We introduce a stepwise algorithm on the basis of the multiple testing viewpoint. Our simulation results for various linear and nonlinear profiles display the superior efficiency of our proposed monitoring procedures over some existing techniques in the literature. To illustrate the implementation of the proposed methods in phase I profile monitoring, we apply the methods on a vertical density profile dataset.

Localising Temperature Risk

On the temperature derivative market, modeling temperature volatility is an important issue for pricing and hedging. In order to apply pricing tools of financial mathematics, one needs to isolate a Gaussian risk factor. A conventional model for temperature dynamics is a stochastic model with seasonality and inter temporal autocorrelation. Empirical work based on seasonality and autocorrelation correction reveals that the obtained residuals are heteroscedastic with a periodic pattern. The object of this research is to estimate this heteroscedastic function so that after scale normalisation a pure standardised Gaussian variable appears. Earlier work investigated this temperature risk in different locations and showed that neither parametric component functions nor a local linear smoother with constant smoothing parameter are flexible enough to generally describe the volatility process well. Therefore, we consider a local adaptive modeling approach to find at each time point, an optimal smoothing parameter to locally estimate the seasonality and volatility. Our approach provides a more flexible and accurate fitting procedure of localised temperature risk process by achieving excellent normal risk factors.

A Confidence Corridor for Sparse Longitudinal Data Curves

Longitudinal data analysis is a central piece of statistics. The data are curves and they are observed at random locations. This makes the construction of a simultaneous confidence corridor (SCC) (confidence band) for the mean function a challenging task on both the theoretical and the practical side. Here we propose a method based on local linear smoothing that is implemented in the sparse (i.e., low number of nonzero coefficients) modelling situation. An SCC is constructed based on recent results obtained in applied probability theory. The precision and performance is demonstrated in a spectrum of simulations and applied to growth curve data. Technically speaking, our paper intensively uses recent insights into extreme value theory that are also employed to construct a shoal of confidence intervals (SCI).

Estimating time-varying treatment switching effects via local linear smoothing and quasi-likelihood

Vascular access complications have been the major cause of excessive morbidity and mortality in the dialysis population. They also account for a large portion of hospitalization for dialysis patients and are a main contributor to the high dialysis care cost. Despite the Fistula First Initiative, the majority of patients initiate dialysis with a central venous catheter which is associated with poor outcomes. In this paper we investigate whether switching from a central venous catheter to an arteriovenous fistula sooner is associated with smaller hospitalization rate. We propose a flexible model for time-varying switching effect while accounting for trend over calendar time, trend over time on dialysis and time-varying effects of covariates. We model all unknown functions nonparametrically using local linear smoothers and estimate them using weighted local quasi-likelihood. We show that the proposed estimators have the desirable large-sample properties and excellent performance in simulations. Application of the proposed method to a real data set indicates that hospitalization rate is smaller when patients switch from a central venous catheter to an arteriovenous fistula sooner. The proposed methods are general which are applicable to other situations with treatment switching.

Generalized semiparametric varying-coefficient model for longitudinal data with applications to adaptive treatment randomizations.

This article investigates a generalized semiparametric varying-coefficient model for longitudinal data that can flexibly model three types of covariate effects: time-constant effects, time-varying effects, and covariate-varying effects. Different link functions can be selected to provide a rich family of models for longitudinal data. The model assumes that the time-varying effects are unspecified functions of time and the covariate-varying effects are parametric functions of an exposure variable specified up to a finite number of unknown parameters. The estimation procedure is developed using local linear smoothing and profile weighted least squares estimation techniques. Hypothesis testing procedures are developed to test the parametric functions of the covariate-varying effects. The asymptotic distributions of the proposed estimators are established. A working formula for bandwidth selection is discussed and examined through simulations. Our simulation study shows that the proposed methods have satisfactory finite sample performance. The proposed methods are applied to the ACTG 244 clinical trial of HIV infected patients being treated with Zidovudine to examine the effects of antiretroviral treatment switching before and after HIV develops the T215Y/F drug resistance mutation. Our analysis shows benefits of treatment switching to the combination therapies as compared to continuing with ZDV monotherapy before and after developing the 215-mutation.

Nonparametric estimation of multivariate multiparameter conditional copulas

Nonparametric estimation of conditional copulas with one parameter has been investigated in Acar et al. (2011). The estimation for multivariate multiparameter conditional copulas, however, has not been considered so far. This paper adopts the local linear smoothing technique and Newton–Raphson method to estimate those copulas. Under some regularity conditions, the asymptotic normality of the estimators is obtained. Simulation work shows the efficiency of the proposed method. As an application, we analyze a life expectancies data set and show that the conditional t copula outperforms the conditional Clayton, Frank and Gumbel copulas.

Localizing Temperature Risk

ABSTRACTOn the temperature derivative market, modeling temperature volatility is an important issue for pricing and hedging. To apply the pricing tools of financial mathematics, one needs to isolate a Gaussian risk factor. A conventional model for temperature dynamics is a stochastic model with seasonality and intertemporal autocorrelation. Empirical work based on seasonality and autocorrelation correction reveals that the obtained residuals are heteroscedastic with a periodic pattern. The object of this research is to estimate this heteroscedastic function so that, after scale normalization, a pure standardized Gaussian variable appears. Earlier works investigated temperature risk in different locations and showed that neither parametric component functions nor a local linear smoother with constant smoothing parameter are flexible enough to generally describe the variance process well. Therefore, we consider a local adaptive modeling approach to find, at each time point, an optimal smoothing parameter to locally estimate the seasonality and volatility. Our approach provides a more flexible and accurate fitting procedure for localized temperature risk by achieving nearly normal risk factors. We also employ our model to forecast the temperaturein different cities and compare it to a model developed in 2005 by Campbell and Diebold. Supplementary materials for this article are available online.

From sparse to dense functional data and beyond

Nonparametric estimation of mean and covariance functions is important in functional data analysis. We investigate the performance of local linear smoothers for both mean and covariance functions with a general weighing scheme, which includes two commonly used schemes, equal weight per observation (OBS), and equal weight per subject (SUBJ), as two special cases. We provide a comprehensive analysis of their asymptotic properties on a unified platform for all types of sampling plan, be it dense, sparse or neither. Three types of asymptotic properties are investigated in this paper: asymptotic normality, $L^{2}$ convergence and uniform convergence. The asymptotic theories are unified on two aspects: (1) the weighing scheme is very general; (2) the magnitude of the number $N_{i}$ of measurements for the $i$th subject relative to the sample size $n$ can vary freely. Based on the relative order of $N_{i}$ to $n$, functional data are partitioned into three types: non-dense, dense and ultra-dense functional data for the OBS and SUBJ schemes. These two weighing schemes are compared both theoretically and numerically. We also propose a new class of weighing schemes in terms of a mixture of the OBS and SUBJ weights, of which theoretical and numerical performances are examined and compared.