Robust nonparametric estimation with missing data

In this paper, under a nonparametric regression model, we introduce two families of robust procedures to estimate the regression function when missing data occur in the response. The first proposal is based on a local M$M$-functional applied to the conditional distribution function estimate adapted to the presence of missing data. The second proposal imputes the missing responses using the local M$M$-smoother based on the observed sample and then estimates the regression function with the completed sample. We show that the robust procedures considered are consistent and asymptotically normally distributed. A robust procedure to select the smoothing parameter is also discussed.     

Semiparametric estimation for weighted average derivatives with responses missing at random

When responses are missing at random, we propose a semiparametric direct estimator for the missing probability and density-weighted average derivatives of a general nonparametric multiple regression function. An estimator for the normalized version of the weighted average derivatives is constructed as well using instrumental variables regression. The proposed estimators are computationally simple and asymptotically normal, and provide a solution to the problem of estimating index coefficients of single-index models with responses missing at random. The developed theory generalizes the method of the density-weighted average derivatives estimation of Powell et al. (1989) for the non-missing data case. Monte Carlo simulation studies are conducted to study the performance of the methods.     

Partially linear single-index model with missing responses at random

This paper considers semiparametric partially linear single-index model with missing responses at random. Imputation approach is developed to estimate the regression coefficients, single-index coefficients and the nonparametric function, respectively. The imputation estimators for the regression coefficients and single-index coefficients are obtained by a stepwise approach. These estimators are shown to be asymptotically normal, and the estimator for the nonparametric function is proved to be asymptotically normal at any fixed point. The bandwidth problem is also considered in this paper, a delete-one cross validation method is used to select the optimal bandwidth. A simulation study is conducted to evaluate the proposed methods.     

Model selection and model averaging for semiparametric partially linear models with missing data

We study model selection and model averaging in semiparametric partially linear models with missing responses. An imputation method is used to estimate the linear regression coefficients and the nonparametric function. We show that the corresponding estimators of the linear regression coefficients are asymptotically normal. Then a focused information criterion and frequentist model average estimators are proposed and their theoretical properties are established. Simulation studies are performed to demonstrate the superiority of the proposed methods over the existing strategies in terms of mean squared error and coverage probability. Finally, the approach is applied to a real data case.     

Estimated estimating equations: semiparametric inference for clustered and longitudinal data

Summary.  We introduce a flexible marginal modelling approach for statistical inference for clustered and longitudinal data under minimal assumptions. This estimated estimating equations approach is semiparametric and the proposed models are fitted by quasi-likelihood regression, where the unknown marginal means are a function of the fixed effects linear predictor with unknown smooth link, and variance–covariance is an unknown smooth function of the marginal means. We propose to estimate the nonparametric link and variance–covariance functions via smoothing methods, whereas the regression parameters are obtained via the estimated estimating equations. These are score equations that contain nonparametric function estimates. The proposed estimated estimating equations approach is motivated by its flexibility and easy implementation. Moreover, if data follow a generalized linear mixed model, with either a specified or an unspecified distribution of random effects and link function, the model proposed emerges as the corresponding marginal (population-average) version and can be used to obtain inference for the fixed effects in the underlying generalized linear mixed model, without the need to specify any other components of this generalized linear mixed model. Among marginal models, the estimated estimating equations approach provides a flexible alternative to modelling with generalized estimating equations. Applications of estimated estimating equations include diagnostics and link selection. The asymptotic distribution of the proposed estimators for the model parameters is derived, enabling statistical inference. Practical illustrations include Poisson modelling of repeated epileptic seizure counts and simulations for clustered binomial responses.     

Quantile regression in functional linear semiparametric model

This paper proposes nonparametric estimation methods for functional linear semiparametric quantile regression, where the conditional quantile of the scalar responses is modelled by both scalar and functional covariates and an additional unknown nonparametric function term. The slope function is estimated using the functional principal component basis and the nonparametric function is approximated by a piecewise polynomial function. The asymptotic distribution of the estimators of slope parameters is derived and the global convergence rate of the quantile estimator of unknown slope function is established under suitable norm. The asymptotic distribution of the estimator of the unknown nonparametric function is also established. Simulation studies are conducted to investigate the finite-sample performance of the proposed estimators. The proposed methodology is demonstrated by analysing a real data from ADHD-200 sample.     

Seemingly unrelated regression tree

Nonparametric seemingly unrelated regression provides a powerful alternative to parametric seemingly unrelated regression for relaxing the linearity assumption. The existing methods are limited, particularly with sharp changes in the relationship between the predictor variables and the corresponding response variable. We propose a new nonparametric method for seemingly unrelated regression, which adopts a tree-structured regression framework, has satisfiable prediction accuracy and interpretability, no restriction on the inclusion of categorical variables, and is less vulnerable to the curse of dimensionality. Moreover, an important feature is constructing a unified tree-structured model for multivariate data, even though the predictor variables corresponding to the response variable are entirely different. This unified model can offer revelatory insights such as underlying economic meaning. We propose the key factors of tree-structured regression, which are an impurity function detecting complex nonlinear relationships between the predictor variables and the response variable, split rule selection with negligible selection bias, and tree size determination solving underfitting and overfitting problems. We demonstrate our proposed method using simulated data and illustrate it using data from the Korea stock exchange sector indices.     

Analysing survey data with incomplete responses by using a method based on empirical likelihood

Summary.  In many surveys, missing response is a common problem. As an example, Zahner, Jacobs, Freeman and Trainor analysed data from a study of child psychopathology in the State of Connecticut, USA. In that study, the response variable, psychopathology, was inferred from questions that were addressed to teachers of the children and was subject to a high level of missingness. However, the missing responses were supplemented by surrogate information that was provided by the parents and/or the primary care providers of the children. In such a situation, it is conceivable that the supplemental information can be used to recover some of the information that has been lost in the cases with missing response. This paper considers a method using empirical likelihood. Empirical likelihood is well known in providing nonparametric inference. But its application has largely been confined to complete-data situations. The method proposed exploits the semiparametric nature of empirical likelihood. The method gives consistent estimates if the cases with non-missing responses form a random sample of the population. In large samples, the method behaves similarly to a regression estimate that is applied to estimating equations. The method is easy to implement with standard statistical packages. In a small sample study, the method was found to give favourable results, when compared with existing methods.     

A comparison study of nonparametric imputation methods

Consider estimation of a population mean of a response variable when the observations are missing at random with respect to the covariate. Two common approaches to imputing the missing values are the nonparametric regression weighting method and the Horvitz-Thompson (HT) inverse weighting approach. The regression approach includes the kernel regression imputation and the nearest neighbor imputation. The HT approach, employing inverse kernel-estimated weights, includes the basic estimator, the ratio estimator and the estimator using inverse kernel-weighted residuals. Asymptotic normality of the nearest neighbor imputation estimators is derived and compared to kernel regression imputation estimator under standard regularity conditions of the regression function and the missing pattern function. A comprehensive simulation study shows that the basic HT estimator is most sensitive to discontinuity in the missing data patterns, and the nearest neighbors estimators can be insensitive to missing data patterns unbalanced with respect to the distribution of the covariate. Empirical studies show that the nearest neighbor imputation method is most effective among these imputation methods for estimating a finite population mean and for classifying the species of the iris flower data.     

基于GMM的缺失数据回归模型的半参数估计

响应变量存在数据缺失的情况广泛出现在社会经济研究中,对响应变量存在数据缺失的回归模型提出了一个在矩估计框架下的单一的半参数估计量,这种估计量保留了参数回归估计量与非参数匹配估计量的特性,从而使得该估计量既能在响应变量被观测的子样本中保持较好的拟合性,又能够降低响应变量未被观测的子样本的估计误差,并且证明了这种估计量是一致、渐进正态估计量。     

A nonparametric hypothesis test for heteroscedasticity

In this paper, a hypothesis test for heteroscedasticity is proposed in a nonparametric regression model. The test statistic, which uses the residuals from a nonparametric fit of the mean function, is based on an adaptation of the well-known Levene's test. Using the recent theory for analysis of variance when the number of factor levels goes to infinity, the asymptotic distribution of the test statistic is established under the null hypothesis of homocedasticity and under local alternatives. Simulations suggest that the proposed test performs well in several situations, especially when the variance is a nonlinear function of the predictor.     

Estimating a Convex Function in Nonparametric Regression

Abstract.  A new nonparametric estimate of a convex regression function is proposed and its stochastic properties are studied. The method starts with an unconstrained estimate of the derivative of the regression function, which is firstly isotonized and then integrated. We prove asymptotic normality of the new estimate and show that it is first order asymptotically equivalent to the initial unconstrained estimate if the regression function is in fact convex. If convexity is not present, the method estimates a convex function whose derivative has the same L ^p -norm as the derivative of the (non-convex) underlying regression function. The finite sample properties of the new estimate are investigated by means of a simulation study and it is compared with a least squares approach of convex estimation. The application of the new method is demonstrated in two data examples.     

Smoothed empirical likelihood analysis of partially linear quantile regression models with missing response variables

In this paper, we consider the estimation and inference of the parameters and the nonparametric part in partially linear quantile regression models with responses that are missing at random. First, we extend the normal approximation (NA)-based methods of Sun (2005) to the missing data case. However, the asymptotic covariance matrices of NA-based methods are difficult to estimate, which complicates inference. To overcome this problem, alternatively, we propose the smoothed empirical likelihood (SEL)-based methods. We define SEL statistics for the parameters and the nonparametric part and demonstrate that the limiting distributions of the statistics are Chi-squared distributions. Accordingly, confidence regions can be obtained without the estimation of the asymptotic covariance matrices. Monte Carlo simulations are conducted to evaluate the performance of the proposed method. Finally, the NA- and SEL-based methods are applied to real data.     

A semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data

We propose a flexible semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data. The model can handle either single cycle or, more generally, multiple consecutive cycle data. The approach models the mean of responses by parametric fixed effects and a smooth nonparametric function for the underlying time effects, and the relationship across the bivariate responses by a bivariate Gaussian random field and a joint distribution of random effects. The proposed model not only can model complicated individual profiles, but also allows for more flexible within-subject and between-response correlations. The fixed effects regression coefficients and the nonparametric time functions are estimated using maximum penalized likelihood, where the resulting estimator for the nonparametric time function is a cubic smoothing spline. The smoothing parameters and variance components are estimated simultaneously using restricted maximum likelihood. Simulation results show that the parameter estimates are close to the true values. The fit of the proposed model on a real bivariate longitudinal dataset of pre-menopausal women also performs well, both for a single cycle analysis and for a multiple consecutive cycle analysis. The Canadian Journal of Statistics 48: 471–498; 2020 © 2020 Statistical Society of Canada     

Strictly monotone and smooth nonparametric regression for two or more variables

The authors propose a new monotone nonparametric estimate for a regression function of two or more variables. Their method consists in applying successively one‐dimensional isotonization procedures on an initial, unconstrained nonparametric regression estimate. In the case of a strictly monotone regression function, they show that the new estimate and the initial one are first‐order asymptotic equivalent; they also establish asymptotic normality of an appropriate standardization of the new estimate. In addition, they show that if the regression function is not monotone in one of its arguments, the new estimate and the initial one have approximately the same L^p‐norm. They illustrate their approach by means of a simulation study, and two data examples are analyzed.     

Model checks for nonparametric regression with missing data: a comparative study

ABSTRACT

This paper analyses the behaviour of the goodness-of-fit tests for regression models. To this end, it uses statistics based on an estimation of the integrated regression function with missing observations either in the response variable or in some of the covariates. It proposes several versions of one empirical process, constructed from a previous estimation, that uses only the complete observations or replaces the missing observations with imputed values. In the case of missing covariates, a link model is used to fill the missing observations with other complete covariates. In all the situations, Bootstrap methodology is used to calibrate the distribution of the test statistics. A broad simulation study compares the different procedures based on empirical regression methodology, with smoothed tests previously studied in the literature. The comparison reflects the effect of the correlation between the covariates in the tests based on the imputed sample for missing covariates. In addition, the paper proposes a computational binning strategy to evaluate the tests based on an empirical process for large data sets. Finally, two applications to real data illustrate the performance of the tests.     

Nonparametric curve estimation with missing data: A general empirical process approach

A general nonparametric imputation procedure, based on kernel regression, is proposed to estimate points as well as set- and function-indexed parameters when the data are missing at random (MAR). The proposed method works by imputing a specific function of a missing value (and not the missing value itself), where the form of this specific function is dictated by the parameter of interest. Both single and multiple imputations are considered. The associated empirical processes provide the right tool to study the uniform convergence properties of the resulting estimators. Our estimators include, as special cases, the imputation estimator of the mean, the estimator of the distribution function proposed by Cheng and Chu [1996. Kernel estimation of distribution functions and quantiles with missing data. Statist. Sinica 6, 63–78], imputation estimators of a marginal density, and imputation estimators of regression functions.     

On the correct regression function (in L2) and its applications when the dimension of the covariate vector is random

We derive the optimal regression function (i.e., the best approximation in the L₂ sense) when the vector of covariates has a random dimension. Furthermore, we consider applications of these results to problems in statistical regression and classification with missing covariates. It will be seen, perhaps surprisingly, that the correct regression function for the case with missing covariates can sometimes perform better than the usual regression function corresponding to the case with no missing covariates. This is because even if some of the covariates are missing, an indicator random variable δ$\delta$, which is always observable, and is equal to 1 if there are no missing values (and 0 otherwise), may have far more information and predictive power about the response variable Y than the missing covariates do. We also propose kernel-based procedures for estimating the correct regression function nonparametrically. As an alternative estimation procedure, we also consider the least-squares method.     

Iterative regularisation in nonparametric instrumental regression

This paper considers the nonparametric regression model with an additive error that is correlated with the explanatory variables. Motivated by empirical studies in epidemiology and economics, it also supposes that valid instrumental variables are observed. However, the estimation of a nonparametric regression function by instrumental variables is an ill-posed linear inverse problem with an unknown but estimable operator. We provide a new estimator of the regression function that is based on projection onto finite dimensional spaces and that includes an iterative regularisation method (the Landweber–Fridman method). The optimal number of iterations and the convergence of the mean square error of the resulting estimator are derived under both strong and weak source conditions. A Monte Carlo exercise shows the impact of some parameters on the estimator and concludes on the reasonable finite sample performance of the new estimator.     

Misclassification of current status data

We describe a simple method for nonparametric estimation of a distribution function based on current status data where observations of current status information are subject to misclassification. Nonparametric maximum likelihood techniques lead to use of a straightforward set of adjustments to the familiar pool-adjacent-violators estimator used when misclassification is assumed absent. The methods consider alternative misclassification models and are extended to regression models for the underlying survival time. The ideas are motivated by and applied to an example on human papilloma virus (HPV) infection status of a sample of women examined in San Francisco.