An F-type test for detecting departure from monotonicity in a functional linear model

When studying associations between a functional covariate and scalar response using a functional linear model (FLM), scientific knowledge may indicate possible monotonicity of the unknown parameter curve. In this context, we propose an F-type test of monotonicity, based on a full versus reduced nested model structure, where the reduced model with monotonically constrained parameter curve is nested within an unconstrained FLM. For estimation under the unconstrained FLM, we consider two approaches: penalised least-squares and linear mixed model effects estimation. We use a smooth then monotonise approach to estimate the reduced model, within the null space of monotone parameter curves. A bootstrap procedure is used to simulate the null distribution of the test statistic. We present a simulation study of the power of the proposed test, and illustrate the test using data from a head and neck cancer study.     

On asymptotic distribution of prediction in functional linear regression

There has been substantial recent attention on problems involving a functional linear regression model with scalar response. Among them, there have been few works dealing with asymptotic distribution of prediction in functional linear regression models. In recent literature, the centeral limit theorem for prediction has been discussed, but the proof and conditions under which the random bias terms for a fixed predictor converge to zero have been ignored so that the impact of these terms on the convergence of the prediction has not been well understood. Clarifying the proof and conditions under which the bias terms converge to zero, we show that the asymptotic distribution of the prediction is normal. Furthermore, we have derived those results related to other terms that already obtained by others, under milder conditions. Finally, we conduct a simulation study to investigate performance of the asymptotic distribution under various parameter settings.     

The historical functional linear model

The authors develop a functional linear model in which the values at time t of a sample of curves yi (t) are explained in a feed‐forward sense by the values of covariate curves xi(s) observed at times s ±.t. They give special attention to the case s ± [t — δ, t], where the lag parameter δ is estimated from the data. They use the finite element method to estimate the bivariate parameter regression function β(s, t), which is defined on the triangular domain s ± t. They apply their model to the problem of predicting the acceleration of the lower lip during speech on the basis of electromyographical recordings from a muscle depressing the lip. They also provide simulation results to guide the calibration of the fitting process.     

Estimation on semi-functional linear errors-in-variables models

Abstract

Semi-functional linear regression models are important in practice. In this paper, their estimation is discussed when function-valued and real-valued random variables are all measured with additive error. By means of functional principal component analysis and kernel smoothing techniques, the estimators of the slope function and the non parametric component are obtained. To account for errors in variables, deconvolution is involved in the construction of a new class of kernel estimators. The convergence rates of the estimators of the unknown slope function and non parametric component are established under suitable norm and conditions. Simulation studies are conducted to illustrate the finite sample performance of our method.     

Generalized linear models with functional predictors

Summary. We present a technique for extending generalized linear models to the situation where some of the predictor variables are observations from a curve or function. The technique is particularly useful when only fragments of each curve have been observed. We demonstrate, on both simulated and real data sets, how this approach can be used to perform linear, logistic and censored regression with functional predictors. In addition, we show how functional principal components can be used to gain insight into the relationship between the response and functional predictors. Finally, we extend the methodology to apply generalized linear models and principal components to standard missing data problems.     

Bootstrap in functional linear regression

We have considered the functional linear model with scalar response and functional explanatory variable. One of the most popular methodologies for estimating the model parameter is based on functional principal components analysis (FPCA). In recent literature, weak convergence for a wide class of FPCA-type estimates has been proved, and consequently asymptotic confidence sets can be built. In this paper, we have proposed an alternative approach in order to obtain pointwise confidence intervals by means of a bootstrap procedure, for which we have obtained its asymptotic validity. Besides, a simulation study allows us to compare the practical behaviour of asymptotic and bootstrap confidence intervals in terms of coverage rates for different sample sizes.     

Estimation and testing for semiparametric mixtures of partially linear models

In this paper, we study the estimation and inference for a class of semiparametric mixtures of partially linear models. We prove that the proposed models are identifiable under mild conditions, and then give a PL–EM algorithm estimation procedure based on profile likelihood. The asymptotic properties for the resulting estimators and the ascent property of the PL–EM algorithm are investigated. Furthermore, we develop a test statistic for testing whether the non parametric component has a linear structure. Monte Carlo simulations and a real data application highlight the interest of the proposed procedures.     

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.     

Estimation for semi-functional linear regression

This paper studies estimation in semi-functional linear regression. A general formulation is used to treat mean regression, median regression, quantile regression and robust mean regression in one setting. The linear slope function is estimated by the functional principal component basis and the nonparametric component is approximated by a B-spline function. The global convergence rates of the estimators of unknown slope function and nonparametric component are established under suitable norm. The convergence rate of the mean-squared prediction error for the proposed estimators is also established. Finite sample properties of our procedures are studied through Monte Carlo simulations. A real data example about Berkeley growth data is used to illustrate our proposed methodology.     

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.     

Semiparametric partially linear regression models for functional data

In the context of longitudinal data analysis, a random function typically represents a subject that is often observed at a small number of time point. For discarding this restricted condition of observation number of each subject, we consider the semiparametric partially linear regression models with mean function x^?β$x^{?}\beta$ + g(z), where x and z   are functional data. The estimations of β$\beta$ and g(z) are presented and some asymptotic results are given. It is shown that the estimator of the parametric component is asymptotically normal. The convergence rate of the estimator of the nonparametric component is also obtained. Here, the observation number of each subject is completely flexible. Some simulation study is conducted to investigate the finite sample performance of the proposed estimators.     

Distribution of the biased hypothesis sum of squares in linear models with missing observations

In a linear model with missing observations, one can substitute algebraic quantities and then minimize the error sum of squares for the augmented model. This gives the correct error sum of squares. But this method does not produce the correct hypothesis sum of squares for testing a linear hypothesis about the parameters. The sum of squares obtained is biased but practitioners still use it. The distribution of this biased sum of squares is derived in this paper and the consequences of using this biased sum of squares on the type I and II errors is examined.     

Missing plot technique in linear models

This paper extends the missing plot substitution technique to the case where the missing observations-cause some previously estimable functions to become non-estimable. It is shown that with appropriate modifications, the usual methods of analysis remain valid. We also obtain necessary and sufficient conditions under which the sum of squares due to a hypothesis can be calculated without “re-estimating” the missing observations     

Smoothing parameter selection for a class of semiparametric linear models

Summary.  Spline-based approaches to non-parametric and semiparametric regression, as well as to regression of scalar outcomes on functional predictors, entail choosing a parameter controlling the extent to which roughness of the fitted function is penalized. We demonstrate that the equations determining two popular methods for smoothing parameter selection, generalized cross-validation and restricted maximum likelihood, share a similar form that allows us to prove several results which are common to both, and to derive a condition under which they yield identical values. These ideas are illustrated by application of functional principal component regression, a method for regressing scalars on functions, to two chemometric data sets.     

A note on functional linear regression

This work focuses on the linear regression model with functional covariate and scalar response. We compare the performance of two (parametric) linear regression estimators and a nonparametric (kernel) estimator via a Monte Carlo simulation study and the analysis of two real data sets. The first linear estimator expands the predictor and the regression weight function in terms of the trigonometric basis, while the second one uses functional principal components. The choice of the regularization degree in the linear estimators is addressed.     

On bootstrap testing inference in cure rate models

Survival models deal with the time until the occurrence of an event of interest. However, in some situations the event may not occur in part of the studied population. The fraction of the population that will never experience the event of interest is generally called cure rate. Models that consider this fact (cure rate models) have been extensively studied in the literature. Hypothesis testing on the parameters of these models can be performed based on likelihood ratio, gradient, score or Wald statistics. Critical values of these tests are obtained through approximations that are valid in large samples and may result in size distortion in small or moderate sample sizes. In this sense, this paper proposes bootstrap corrections to the four mentioned tests and bootstrap Bartlett correction for the likelihood ratio statistic in the Weibull promotion time model. Besides, we present an algorithm for bootstrap resampling when the data presents cure fraction and right censoring time (random and non-informative). Simulation studies are conducted to compare the finite sample performances of the corrected tests. The numerical evidence favours the corrected tests we propose. We also present an application in an actual data set.     

Estimation and variable selection for partially functional linear models

In this paper, a new estimation procedure based on composite quantile regression and functional principal component analysis (PCA) method is proposed for the partially functional linear regression models (PFLRMs). The proposed estimation method can simultaneously estimate both the parametric regression coefficients and functional coefficient components without specification of the error distributions. The proposed estimation method is shown to be more efficient empirically for non-normal random error, especially for Cauchy error, and almost as efficient for normal random errors. Furthermore, based on the proposed estimation procedure, we use the penalized composite quantile regression method to study variable selection for parametric part in the PFLRMs. Under certain regularity conditions, consistency, asymptotic normality, and Oracle property of the resulting estimators are derived. Simulation studies and a real data analysis are conducted to assess the finite sample performance of the proposed methods.     

Statistical estimation for partially linear error-in-variable models with error-prone covariates

Motivated by a heart disease data, we propose a new partially linear error-in-variable models with error-prone covariates, in which mismeasured covariate appears in the noparametric part and the covariates in the parametric part are not observed, but ancillary variables are available. In this case, we first calibrate the linear covariates, and then use the least-square method and the local linear method to estimate parametric and nonparametric components. Also, under certain conditions the asymptotic distributions of proposed estimates are obtained. Simulated and real examples are conducted to illustrate our proposed methodology.