Optimal estimation in functional linear regression for sparse noise‐contaminated data

In this article, we propose a novel approach to fit a functional linear regression in which both the response and the predictor are functions. We consider the case where the response and the predictor processes are both sparsely sampled at random time points and are contaminated with random errors. In addition, the random times are allowed to be different for the measurements of the predictor and the response functions. The aforementioned situation often occurs in longitudinal data settings. To estimate the covariance and the cross‐covariance functions, we use a regularization method over a reproducing kernel Hilbert space. The estimate of the cross‐covariance function is used to obtain estimates of the regression coefficient function and of the functional singular components. We derive the convergence rates of the proposed cross‐covariance, the regression coefficient, and the singular component function estimators. Furthermore, we show that, under some regularity conditions, the estimator of the coefficient function has a minimax optimal rate. We conduct a simulation study and demonstrate merits of the proposed method by comparing it to some other existing methods in the literature. We illustrate the method by an example of an application to a real‐world air quality dataset. The Canadian Journal of Statistics 47: 524–559; 2019 © 2019 Statistical Society of Canada     

Using ranked set sampling with binary outcomes in cluster randomized designs

We study the use of ranked set sampling (RSS) with binary outcomes in cluster-randomized designs (CRDs), where a generalized linear mixed model (GLMM) is used to model the hierarchical data structure involved. Under the GLMM-based framework, we propose three different approaches to estimate the treatment effect, including the nonparametric (NP), maximum likelihood (ML) and pseudo likelihood (PL) estimators. We investigate their asymptotic properties and examine their finite-sample performance via simulation. Based on these three RSS estimators, we further develop procedures for testing the existence of the treatment effect. We examine the power and size of our proposed RSS tests and compare them with existing tests based on simple random sampling (SRS). All the proposed RSS estimation and test methods are illustrated with two data examples, one for rare events and the other for non-extreme events. Throughout our investigations, we also consider the possible effect of imperfect ranking. Among the proposed methods, we provide recommendations on whether to use RSS rather than SRS with binary outcomes in CRDs and, if yes, when to use which RSS method. The Canadian Journal of Statistics 48: 342–365; 2020 © 2019 Statistical Society of Canada     

A new distribution-free k-sample test: Analysis of kernel density functionals

A novel distribution-free k-sample test of differences in location shifts based on the analysis of kernel density functional estimation is introduced and studied. The proposed test parallels one-way analysis of variance and the Kruskal–Wallis (KW) test aiming at testing locations of unknown distributions. In contrast to the rank (score)-transformed non-parametric approach, such as the KW test, the proposed F-test uses the measurement responses along with well-known kernel density estimation (KDE) to estimate the locations and construct the test statistic. A practical optimal bandwidth selection procedure is also provided. Our simulation studies and real data example indicate that the proposed analysis of kernel density functional estimate (ANDFE) test is superior to existing competitors for fat-tailed or heavy-tailed distributions when the k groups differ mainly in location rather than shape, especially with unbalanced data. ANDFE is also highly recommended when it is unclear whether test groups differ mainly in shape or location. The Canadian Journal of Statistics 48: 167–186; 2020 © 2019 Statistical Society of Canada     

Functional Partial Linear Single‐index Model

This paper deals with the problem of predicting the real‐valued response variable using explanatory variables containing both multivariate random variable and random curve. The proposed functional partial linear single‐index model treats the multivariate random variable as linear part and the random curve as functional single‐index part, respectively. To estimate the non‐parametric link function, the functional single‐index and the parameters in the linear part, a two‐stage estimation procedure is proposed. Compared with existing semi‐parametric methods, the proposed approach requires no initial estimation and iteration. Asymptotical properties are established for both the parameters in the linear part and the functional single‐index. The convergence rate for the non‐parametric link function is also given. In addition, asymptotical normality of the error variance is obtained that facilitates the construction of confidence region and hypothesis testing for the unknown parameter. Numerical experiments including simulation studies and a real‐data analysis are conducted to evaluate the empirical performance of the proposed method.     

Partially functional linear varying coefficient model

In this paper, we introduce a new partially functional linear varying coefficient model, where the response is a scalar and some of the covariates are functional. By means of functional principal components analysis and local linear smoothing techniques, we obtain the estimators of coefficient functions of both function-valued variable and real-valued variables. Then the rates of convergence of the proposed estimators and the mean squared prediction error are established under some regularity conditions. Moreover, we develop a hypothesis test for the model and employ the bootstrap procedure to evaluate the null distribution of test statistic and the p-value of the test. At last, we illustrate the finite sample performance of our methods with some simulation studies and a real data application.     

Mixtures of Linear Regression with Measurement Errors

Existing research on mixtures of regression models are limited to directly observed predictors. The estimation of mixtures of regression for measurement error data imposes challenges for statisticians. For linear regression models with measurement error data, the naive ordinary least squares method, which directly substitutes the observed surrogates for the unobserved error-prone variables, yields an inconsistent estimate for the regression coefficients. The same inconsistency also happens to the naive mixtures of regression estimate, which is based on the traditional maximum likelihood estimator and simply ignores the measurement error. To solve this inconsistency, we propose to use the deconvolution method to estimate the mixture likelihood of the observed surrogates. Then our proposed estimate is found by maximizing the estimated mixture likelihood. In addition, a generalized EM algorithm is also developed to find the estimate. The simulation results demonstrate that the proposed estimation procedures work well and perform much better than the naive estimates.     

Optimal design for classification of functional data

We study the design problem for the optimal classification of functional data. The goal is to select sampling time points so that functional data observed at these time points can be classified accurately. We propose optimal designs that are applicable to either dense or sparse functional data. Using linear discriminant analysis, we formulate our design objectives as explicit functions of the sampling points. We study the theoretical properties of the proposed design objectives and provide a practical implementation. The performance of the proposed design is evaluated through simulations and real data applications. The Canadian Journal of Statistics 48: 285–307; 2020 © 2019 Statistical Society of Canada     

Local and omnibus goodness-of-fit tests in classical measurement error models

We consider functional measurement error models, i.e. models where covariates are measured with error and yet no distributional assumptions are made about the mismeasured variable. We propose and study a score-type local test and an orthogonal series-based, omnibus goodness-of-fit test in this context, where no likelihood function is available or calculated-i.e. all the tests are proposed in the semiparametric model framework. We demonstrate that our tests have optimality properties and computational advantages that are similar to those of the classical score tests in the parametric model framework. The test procedures are applicable to several semiparametric extensions of measurement error models, including when the measurement error distribution is estimated non-parametrically as well as for generalized partially linear models. The performance of the local score-type and omnibus goodness-of-fit tests is demonstrated through simulation studies and analysis of a nutrition data set.     

Empirical likelihood for nonlinear regression models with nonignorable missing responses

This article develops three empirical likelihood (EL) approaches to estimate parameters in nonlinear regression models in the presence of nonignorable missing responses. These are based on the inverse probability weighted (IPW) method, the augmented IPW (AIPW) method and the imputation technique. A logistic regression model is adopted to specify the propensity score. Maximum likelihood estimation is used to estimate parameters in the propensity score by combining the idea of importance sampling and imputing estimating equations. Under some regularity conditions, we obtain the asymptotic properties of the maximum EL estimators of these unknown parameters. Simulation studies are conducted to investigate the finite sample performance of our proposed estimation procedures. Empirical results provide evidence that the AIPW procedure exhibits better performance than the other two procedures. Data from a survey conducted in 2002 are used to illustrate the proposed estimation procedure. The Canadian Journal of Statistics 48: 386–416; 2020 © 2020 Statistical Society of Canada     

Linear mode regression with covariate measurement error

We consider estimating the mode of a response given an error‐prone covariate. It is shown that ignoring measurement error typically leads to inconsistent inference for the conditional mode of the response given the true covariate, as well as misleading inference for regression coefficients in the conditional mode model. To account for measurement error, we first employ the Monte Carlo corrected score method (Novick & Stefanski, 2002) to obtain an unbiased score function based on which the regression coefficients can be estimated consistently. To relax the normality assumption on measurement error this method requires, we propose another method where deconvoluting kernels are used to construct an objective function that is maximized to obtain consistent estimators of the regression coefficients. Besides rigorous investigation on asymptotic properties of the new estimators, we study their finite sample performance via extensive simulation experiments, and find that the proposed methods substantially outperform a naive inference method that ignores measurement error. The Canadian Journal of Statistics 47: 262–280; 2019 © 2019 Statistical Society of Canada     

A Note on Generalized Functional Linear Model and Its Application

Motivated by a biomarker study for colorectal neoplasia, we consider generalized functional linear models where the functional predictors are measured with errors at discrete design points. Assuming that the true functional predictor and the slope function are smooth, we investigate a two-step estimating procedure where both the true functional predictor and the slope function are estimated through spline smoothing. The operating characteristics of the proposed method are derived; the usefulness of the proposed method is illustrated by a simulation study as well as data analysis for the motivating colorectal neoplasia study.     

Estimating prediction error for complex samples

With a growing interest in using non-representative samples to train prediction models for numerous outcomes it is necessary to account for the sampling design that gives rise to the data in order to assess the generalized predictive utility of a proposed prediction rule. After learning a prediction rule based on a non-uniform sample, it is of interest to estimate the rule's error rate when applied to unobserved members of the population. Efron (1986) proposed a general class of covariance penalty inflated prediction error estimators that assume the available training data are representative of the target population for which the prediction rule is to be applied. We extend Efron's estimator to the complex sample context by incorporating Horvitz–Thompson sampling weights and show that it is consistent for the true generalization error rate when applied to the underlying superpopulation. The resulting Horvitz–Thompson–Efron estimator is equivalent to dAIC, a recent extension of Akaike's information criteria to survey sampling data, but is more widely applicable. The proposed methodology is assessed with simulations and is applied to models predicting renal function obtained from the large-scale National Health and Nutrition Examination Study survey. The Canadian Journal of Statistics 48: 204–221; 2020 © 2019 Statistical Society of Canada     

Local quadratic estimation of the curvature in a functional single index model

The nonlinear responses of species to environmental variability can play an important role in the maintenance of ecological diversity. Nonetheless, many models use parametric nonlinear terms which pre-determine the ecological conclusions. Motivated by this concern, we study the estimate of the second derivative (curvature) of the link function in a functional single index model. Since the coefficient function and the link function are both unknown, the estimate is expressed as a nested optimization. We first estimate the coefficient function by minimizing squared error where the link function is estimated with a Nadaraya-Watson estimator for each candidate coefficient function. The first and second derivatives of the link function are then estimated via local-quadratic regression using the estimated coefficient function. In this paper, we derive a convergence rate for the curvature of the nonlinear response. In addition, we prove that the argument of the linear predictor can be estimated root-n consistently. However, practical implementation of the method requires solving a nonlinear optimization problem, and our results show that the estimates of the link function and the coefficient function are quite sensitive to the choices of starting values.     

Classical testing in functional linear models

We extend four tests common in classical regression – Wald, score, likelihood ratio and F tests – to functional linear regression, for testing the null hypothesis, that there is no association between a scalar response and a functional covariate. Using functional principal component analysis, we re-express the functional linear model as a standard linear model, where the effect of the functional covariate can be approximated by a finite linear combination of the functional principal component scores. In this setting, we consider application of the four traditional tests. The proposed testing procedures are investigated theoretically for densely observed functional covariates when the number of principal components diverges. Using the theoretical distribution of the tests under the alternative hypothesis, we develop a procedure for sample size calculation in the context of functional linear regression. The four tests are further compared numerically for both densely and sparsely observed noisy functional data in simulation experiments and using two real data applications.     

Incomplete covariates data in generalized linear models

We consider regression analysis when part of covariates are incomplete in generalized linear models. The incomplete covariates could be due to measurement error or missing for some study subjects. We assume there exists a validation sample in which the data is complete and is a simple random subsample from the whole sample. Based on the idea of projection-solution method in Heyde (1997, Quasi-Likelihood and its Applications: A General Approach to Optimal Parameter Estimation. Springer, New York), a class of estimating functions is proposed to estimate the regression coefficients through the whole data. This method does not need to specify a correct parametric model for the incomplete covariates to yield a consistent estimate, and avoids the ‘curse of dimensionality’ encountered in the existing semiparametric method. Simulation results shows that the finite sample performance and efficiency property of the proposed estimates are satisfactory. Also this approach is computationally convenient hence can be applied to daily data analysis.     

Penalized contrast estimation in functional linear models with circular data

Our aim is to estimate the unknown slope function in the functional linear model when the response Y is real and the random function X is a second-order stationary and periodic process. We obtain our estimator by minimizing a standard (and very simple) mean-square contrast on linear finite dimensional spaces spanned by trigonometric bases. Our approach provides a penalization procedure which allows to automatically select the adequate dimension, in a non-asymptotic point of view. In fact, we can show that our penalized estimator reaches the optimal (minimax) rate of convergence in the sense of the prediction error. We complete the theoretical results by a simulation study and a real example that illustrates how the procedure works in practice.     

Composite quantile regression for varying-coefficient single-index models

ABSTRACT

The varying-coefficient single-index model (VCSIM) is a very general and flexible tool for exploring the relationship between a response variable and a set of predictors. Popular special cases include single-index models and varying-coefficient models. In order to estimate the index-coefficient and the non parametric varying-coefficients in the VCSIM, we propose a two-stage composite quantile regression estimation procedure, which integrates the local linear smoothing method and the information of quantile regressions at a number of conditional quantiles of the response variable. We establish the asymptotic properties of the proposed estimators for the index-coefficient and varying-coefficients when the error is heterogeneous. When compared with the existing mean-regression-based estimation method, our simulation results indicate that our proposed method has comparable performance for normal error and is more robust for error with outliers or heavy tail. We illustrate our methodologies with a real example.     

Multivariate functional response low‐rank regression with an application to brain imaging data

We propose a multivariate functional response low‐rank regression model with possible high‐dimensional functional responses and scalar covariates. By expanding the slope functions on a set of sieve bases, we reconstruct the basis coefficients as a matrix. To estimate these coefficients, we propose an efficient procedure using nuclear norm regularization. We also derive error bounds for our estimates and evaluate our method using simulations. We further apply our method to the Human Connectome Project neuroimaging data to predict cortical surface motor task‐evoked functional magnetic resonance imaging signals using various clinical covariates to illustrate the usefulness of our results.     

Semiparametric median residual life model and inference

For randomly censored data, the authors propose a general class of semiparametric median residual life models. They incorporate covariates in a generalized linear form while leaving the baseline median residual life function completely unspecified. Despite the non‐identifiability of the survival function for a given median residual life function, a simple and natural procedure is proposed to estimate the regression parameters and the baseline median residual life function. The authors derive the asymptotic properties for the estimators, and demonstrate the numerical performance of the proposed method through simulation studies. The median residual life model can be easily generalized to model other quantiles, and the estimation method can also be applied to the mean residual life model. The Canadian Journal of Statistics 38: 665–679; 2010 © 2010 Statistical Society of Canada     

Estimation in autoregressive models with surrogate data and validation data

Time-series data are often subject to measurement error, usually the result of needing to estimate the variable of interest. Generally, however, the relationship between the surrogate variables and the true variables can be rather complicated compared to the classical additive error structure usually assumed. In this article, we address the estimation of the parameters in autoregressive models in the presence of function measurement errors. We first develop a parameter estimation method with the help of validation data; this estimation method does not depend on functional form and the distribution of the measurement error. The proposed estimator is proved to be consistent. Moreover, the asymptotic representation and the asymptotic normality of the estimator are also derived, respectively. Simulation results indicate that the proposed method works well for practical situation.