Functional measurement error in functional regression

Measurement error is an important problem that has not been studied very well in the context of functional data analysis. To the best of our knowledge, there are no existing methods that address the presence of functional measurement errors in generalized functional linear models. In this article, a novel approach is proposed to estimate the slope function in the presence of measurement error in the generalized functional linear model with a scalar response. This work significantly advances the existing conditional score method to accommodate the case where both the measurement error and independent variables lie in infinite dimensional spaces. Asymptotic results are established for the proposed estimate, and its behaviour is studied via simulations, where the response is continuous or binary. Analysis of Canadian Weather data highlights the practical utility of our method. The Canadian Journal of Statistics 48: 238–258; 2020 © 2020 Statistical Society of Canada     

Varying-coefficient single-index measurement error model

This paper proposes a varying-coefficient single-index measurement error model, which consists of measurement error in the index covariates. We combine the simulation-extrapolation technique, the local linear regression and the weighted least-squares method to estimate the unknowns of the current model, and develop the asymptotic properties of the resulting estimators under some conditions. A simulation study is conducted to evaluate the proposed methodology, and a real example is also studied to illustrate our given methodology.     

Consistent estimation of regression coefficients in ultrastructural measurement error model using stochastic prior information

The problem of consistent estimation of regression coefficients in a multivariate linear ultrastructural measurement error model is considered in this article when some additional information on regression coefficients is available a priori. Such additional information is expressible in the form of stochastic linear restrictions. Utilizing stochastic restrictions given a priori, some methodologies are presented to obtain the consistent estimators of regression coefficients under two types of additional information separately, viz., covariance matrix of measurement errors and reliability matrix associated with explanatory variables. The measurement errors are assumed to be not necessarily normally distributed. The asymptotic properties of the proposed estimators are derived and analyzed analytically as well as numerically through a Monte Carlo simulation experiment.     

SIMEX estimation for single-index model with covariate measurement error

AStA Advances in Statistical Analysis - 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...     

Generalized confidence interval for the slope in linear measurement error model

A generalized confidence interval for the slope parameter in linear measurement error model is proposed in this article, which is based on the relation between the slope of classical regression model and the measurement error model. The performance of the confidence interval estimation procedure is studied numerically through Monte Carlo simulation in terms of coverage probability and expected length.     

Sparsity identification for high-dimensional partially linear model with measurement error

In this article, we studied the identification of significant predictors in partially linear model in which some regressors are contaminated with random errors. Moreover, the dimension of parametric component is divergent and the regression coefficients are sparse. We applied difference technique to remove the nonparametric component for circumventing the selection of bandwidth, and constructed a bias-corrected shrinking estimator for the coefficient by using smoothly clipped absolute deviation (SCAD) penalty. Then, we derived the estimating and selecting consistency and established the asymptotic distribution for the identified significant estimators. Finally, Monte Carlo studies illustrate the performance of our approach.     

Least absolute relative error estimation for functional quadratic multiplicative model

ABSTRACT

As an alternative to the functional quadratic model due to Yao and Müller (2010 Yao, F., Müller, H.-G. (2010). Functional quadratic regression. Biometrika 97:49–64.[Crossref], [Web of Science ®] , [Google Scholar]), we consider a functional quadratic multiplicative model. This multiplicative model provides a useful alternative when the relative error is considered for analyzing data with positive responses. The existing work for functional models are mainly based on absolute errors. The commonly used least squares criterion is just such an example. In many practical applications, however, people concern on the size of relative error rather than that of error itself. Therefore, the estimation procedure based on least absolute relative errors, which is proposed by Chen et al. (2010 Chen, K., Guo, S., Lin, Y., Ying, Z. (2010). Least absolute relative error estimation. J. Am. Stat. Assoc. 105:1104–1112.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) for the linear multiplicative model, is developed for functional quadratic multiplicative model. The asymptotic behaviors of the proposed estimators are established. Some simulation studies show that the estimation procedure has good prediction performance. Moreover, a real data set is analyzed for illustrating the proposed methods.     

The proportional hazards model with covariate measurement error

The proportional hazards regression model is commonly used to evaluate the relationship between survival and covariates. Covariates are frequently measured with error. Substituting mismeasured values for the true covariates leads to biased estimation. Hu et al. (Biometrics 88 (1998) 447) have proposed to base estimation in the proportional hazards model with covariate measurement error on a joint likelihood for survival and the covariate variable. Nonparametric maximum likelihood estimation (NPMLE) was used and simulations were conducted to assess the asymptotic validity of this approach. In this paper, we derive a rigorous proof of asymptotic normality of the NPML estimators.     

Robust inference in an heteroscedastic measurement error model

In this paper we deal with robust inference in heteroscedastic measurement error models. Rather than the normal distribution, we postulate a Student t distribution for the observed variables. Maximum likelihood estimates are computed numerically. Consistent estimation of the asymptotic covariance matrices of the maximum likelihood and generalized least squares estimators is also discussed. Three test statistics are proposed for testing hypotheses of interest with the asymptotic chi-square distribution which guarantees correct asymptotic significance levels. Results of simulations and an application to a real data set are also reported.     

Replicated measurement error model under exact linear restrictions

We consider a replicated ultrastructural measurement error regression model where predictor variables are observed with error. It is assumed that some prior information regarding the regression coefficients is available in the form of exact linear restrictions. Three classes of estimators of regression coefficients are proposed. These estimators are shown to be consistent as well as satisfying the given restrictions. The asymptotic properties of unrestricted as well as restricted estimators are studied without imposing any distributional assumption on any random component of the model. A Monte Carlo simulations study is performed to assess the effect of sample size, replicates and non-normality on the estimators.     

Statistical inference for a semiparametric measurement error regression model with heteroscedastic errors

Efficient inference for regression models requires that the heteroscedasticity be taken into account. We consider statistical inference under heteroscedasticity in a semiparametric measurement error regression model, in which some covariates are measured with errors. This paper has multiple components. First, we propose a new method for testing the heteroscedasticity. The advantages of the proposed method over the existing ones are that it does not need any nonparametric estimation and does not involve any mismeasured variables. Second, we propose a new two-step estimator for the error variances if there is heteroscedasticity. Finally, we propose a weighted estimating equation-based estimator (WEEBE) for the regression coefficients and establish its asymptotic properties. Compared with existing estimators, the proposed WEEBE is asymptotically more efficient, avoids undersmoothing the regressor functions and requires less restrictions on the observed regressors. Simulation studies show that the proposed test procedure and estimators have nice finite sample performance. A real data set is used to illustrate the utility of our proposed methods.     

Variable selection for partially linear proportional hazards model with covariate measurement error

In survival analysis, we may encounter the following three problems: nonlinear covariate effect, variable selection and measurement error. Existing studies only address one or two of these problems. The goal of this study is to fill the knowledge gap and develop a novel approach to simultaneously address all three problems. Specifically, a partially time-varying coefficient proportional hazards model is proposed to more flexibly describe covariate effects. Corrected score and conditional score approaches are employed to accommodate potential measurement error. For the selection of relevant variables and regularised estimation, a penalisation approach is adopted. It is shown that the proposed approach has satisfactory asymptotic properties. It can be effectively realised using an iterative algorithm. The performance of the proposed approach is assessed via simulation studies and further illustrated by application to data from an AIDS clinical trial.     

Bayesian analysis for a skew extension of the multivariate null intercept measurement error model

Skew-normal distribution is a class of distributions that includes the normal distributions as a special case. In this paper, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis in a multivariate, null intercept, measurement error model [R. Aoki, H. Bolfarine, J.A. Achcar, and D. Leão Pinto Jr, Bayesian analysis of a multivariate null intercept error-in-variables regression model, J. Biopharm. Stat. 13(4) (2003b), pp. 763–771] where the unobserved value of the covariate (latent variable) follows a skew-normal distribution. The results and methods are applied to a real dental clinical trial presented in [A. Hadgu and G. Koch, Application of generalized estimating equations to a dental randomized clinical trial, J. Biopharm. Stat. 9 (1999), pp. 161–178].     

Instrumental variable estimation in a probit measurement error model

Probit regression is studied when normally distributed covariates are subject to normally distributed measurement errors. Under the assumption that surrogate instrumental variables are available, the parameters in the probit model are shown to be identified. The maximum likelihood estimator and an easily computed two-stage estimator are derived and studied. The two-stage estimator is shown to be asymptotically efficient. Simulation results complement the theory and provide evidence of robustness to the normality assumptions.     

Robust replicated heteroscedastic measurement error model using heavy-tailed distribution

Heteroscedastic measurement error models are widely used in epidemiological, analytical chemistry, and other research areas. In this article, we propose a heteroscedastic measurement error model for replicated data under scale mixtures of normal distributions with/without equation error, which covers unpair and/or unequal replication cases. We obtain iterative formulas of maximum likelihood estimations via EM algorithm, and provide closed forms of asymptotic variances of the estimators. Simulation studies and a real data application are reported to investigate the effective and robust performances of the model and estimates.     

Pseudo-empirical Bayes estimation of small area means under a nested error linear regression model with functional measurement errors

Small area estimation is studied under a nested error linear regression model with area level covariate subject to measurement error. Ghosh and Sinha (2007) obtained a pseudo-Bayes (PB) predictor of a small area mean and a corresponding pseudo-empirical Bayes (PEB) predictor, using the sample means of the observed covariate values to estimate the true covariate values. In this paper, we first derive an efficient PB predictor by using all the available data to estimate true covariate values. We then obtain a corresponding PEB predictor and show that it is asymptotically “optimal”. In addition, we employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator. Our results show that the proposed PEB predictor can lead to significant gain in efficiency over the previously proposed PEB predictor. Area level models are also studied.     

A variance shift model for detection of outliers in the linear mixed measurement error models

ABSTRACT

In this paper, we extend a variance shift model, previously considered in the linear mixed models, to the linear mixed measurement error models using the corrected likelihood of Nakamura (1990 Nakamura, T. (1990). Corrected score function for errors in variables models: methodology and application to generalized linear models. Biometrika 77:127–137.[Crossref], [Web of Science ®] , [Google Scholar]). This model assumes that a single outlier arises from an observation with inflated variance. We derive the score test and the analogue of the likelihood ratio test, to assess whether the ith observation has inflated variance. A parametric bootstrap procedure is implemented to obtain empirical distributions of the test statistics. Finally, results of a simulation study and an example of real data are presented to illustrate the performance of proposed tests.     

A corrected likelihood method for the proportional hazards model with covariates subject to measurement error

There has been extensive interest in discussing inference methods for survival data when some covariates are subject to measurement error. It is known that standard inferential procedures produce biased estimation if measurement error is not taken into account. With the Cox proportional hazards model a number of methods have been proposed to correct bias induced by measurement error, where the attention centers on utilizing the partial likelihood function. It is also of interest to understand the impact on estimation of the baseline hazard function in settings with mismeasured covariates. In this paper we employ a weakly parametric form for the baseline hazard function and propose simple unbiased estimating functions for estimation of parameters. The proposed method is easy to implement and it reveals the connection between the naive method ignoring measurement error and the corrected method with measurement error accounted for. Simulation studies are carried out to evaluate the performance of the estimators as well as the impact of ignoring measurement error in covariates. As an illustration we apply the proposed methods to analyze a data set arising from the Busselton Health Study [Knuiman, M.W., Cullent, K.J., Bulsara, M.K., Welborn, T.A., Hobbs, M.S.T., 1994. Mortality trends, 1965 to 1989, in Busselton, the site of repeated health surveys and interventions. Austral. J. Public Health 18, 129–135].