Functional Measurement Error Models Research Articles

Conditional likelihood inference in a heteroscedastic functional measurement error model

In this paper, we deal with inference about the structural parameters in a heteroscedastic functional measurement error models under the normal distribution assumption. Given a minimal sufficient statistic for the incidental parameters, the conditional maximum likelihood (CML) approach is used. We show that CML estimators have explicit expressions and their sampling distribution is exact. We also show that the classical test statistics to test hypotheses of interest coincide and have exact distributions. We apply the statistical inference tools developed to a data set on comparison of measurement methods.

Optimal designs for homoscedastic functional polynomial measurement error models

This paper considers the construction of optimal designs for homoscedastic functional polynomial measurement error models. The general equivalence theorems are given to check the optimality of a given design, based on the locally and Bayesian D-optimality criteria. The explicit characterizations of the locally and Bayesian D-optimal designs are provided. The results are illustrated by numerical analysis for a quadratic polynomial measurement error model. Numerical results show that the error-variances ratio and the model parameter are the important factors for the both optimal designs. Moreover, it is shown that the Bayesian D-optimal design is more robust and effective compared with the locally D-optimal design, if the error-variances ratio or the model parameter is misspecified.

Multivariable Bayesian curve-fitting under functional measurement error model

Multivariable Bayesian curve-fitting under functional measurement error model

Asymptotic normality of total least squares estimator in a multivariate errors-in-variables model $AX = B$

We consider a multivariate functional measurement error model $AX\approx B$. The errors in $[A,B]$ are uncorrelated, row-wise independent, and have equal (unknown) variances. We study the total least squares estimator of $X$, which, in the case of normal errors, coincides with the maximum likelihood one. We give conditions for asymptotic normality of the estimator when the number of rows in $A$ is increasing. Under mild assumptions, the covariance structure of the limit Gaussian random matrix is nonsingular. For normal errors, the results can be used to construct an asymptotic confidence interval for a linear functional of $X$.

A test for the slope in the functional measurement error model

There are two measurement error models in linear regression, the structural and the functional. Theoretical investigations and applications are concentrated on the structural model with jointly normally and identically distributed observations because there is no test for the slope in the functional model so far. This gap will be closed here for the model with one independent variable. Furthermore it is stated that the functional model is a natural extension of the classical linear regression model with one independent variable if there are errors of measurement in both variables. Moreover it is not postulated in the functional model that the expectations are equal. So the functional model is more realistic than the structural.

Bayesian curve-fitting with radial basis functions under functional measurement error model

Abstract This article presents Bayesian approach to regression splines with knots on a grid ofequally spaced sample quantiles of the independent variables under functional measure-ment error model. We consider small area model by using penalized splines of non-linearpattern. Speci cally, in a basis functions of the regression spline, we use radial basisfunctions. To t the model and estimate parameters we suggest a hierarchical Bayesianframework using Markov Chain Monte Carlo methodology. Furthermore, we illustratethe method in an application data. We check the convergence by a potential scale re-duction factor and we use the posterior predictive p value and the mean logarithmicconditional predictive ordinate to compar models.Keywords: Functional, hierarchical Bayes, measurement error, radial basis, semipara-metric. 1. Introduction We developed the semiparametric small area models with measurement errors models inour previous paper (Hwang and Kim, 2010). Speci cally, we considered small area model byusing penalized splines of non-linear pattern based on truncated polynomial basis functionsand knots on a grid of equally spaced sample quantiles under functional measurement errormodel. Measurement error modeling is also related to Goo and Kim (2013).The truncated polynomial basis functions (TPBF) is simple, but not always numericallystable when the number of knots is large and the smoothing parameter close to zero. In thiscase the computation has to be organized carefully and numerically superior alternativesare available, like B-splines and radial basis functions (Ruppert et al., 2003).The objective of this article is to develop alternative estimators of small area means byusing radial basis functions (RBF) with functional measurement error model. RBF is de nedwith degree pfrom Ruppert et al. (2003) as follows.1;x; ;x

Statistical calibration of qRT-PCR, microarray and RNA-Seq gene expression data with measurement error models

The accurate quantification of gene expression levels is crucial for transcriptome study. Microarray platforms are commonly used for simultaneously interrogating thousands of genes in the past decade, and recently RNA-Seq has emerged as a promising alternative. The gene expression measurements obtained by microarray and RNA-Seq are, however, subject to various measurement errors. A third platform called qRT-PCR is acknowledged to provide more accurate quantification of gene expression levels than microarray and RNA-Seq, but it has limited throughput capacity. In this article, we propose to use a system of functional measurement error models to model gene expression measurements and calibrate the microarray and RNA-Seq platforms with qRT-PCR. Based on the system, a two-step approach was developed to estimate the biases and error variance components of the three platforms and calculate calibrated estimates of gene expression levels. The estimated biases and variance components shed light on the relative strengths and weaknesses of the three platforms and the calibrated estimates provide a more accurate and consistent quantification of gene expression levels. Theoretical and simulation studies were conducted to establish the properties of those estimates. The system was applied to analyze two gene expression data sets from the Microarray Quality Control (MAQC) and Sequencing Quality Control (SEQC) projects.

Influence measures on corrected score estimators in functional heteroscedastic measurement error models

This paper deals with the local influence assessment of the effects of minor perturbations of data on corrected score estimators in the functional heteroscedastic measurement error models with known variances. By extending to the context of measurement error models the differential-geometrical framework proposed by Zhu et al. [H.T. Zhu, J.G. Ibrahim, S. Lee, H. Zhang, Perturbation selection and influence measures in local influence analysis, The Annals of Statistics 35 (2007) 2565–2588], an n-dimensional Riemannian manifold is defined. The associated metric tensor is utilized for the selection of appropriate perturbation schemes. The Levi-Civita connection and first and second derivatives of the corrected score estimators are used to construct influence measures. Simple formulas are obtained under different perturbation schemes. A comparison with the slope and curvature based diagnostics defined from the surface of the corrected score estimators formed by perturbation is included. A real data application and a simulated example illustrate the performance of the proposed diagnostics.

Saddlepoint Test in Measurement Error Models

We develop second-order hypothesis testing procedures in functional measurement error models for small or moderate sample sizes, where the classical first-order asymptotic analysis often fails to provide accurate results. In functional models no distributional assumptions are made on the unobservable covariates and this leads to semiparametric models. Our testing procedure is derived using saddlepoint techniques and is based on an empirical distribution estimation subject to the null hypothesis constraints, in combination with a set of estimating equations which avoid a distribution approximation. The validity of the method is proved in theorems for both simple and composite hypothesis tests, and is demonstrated through simulation and a farm size data analysis.

Generalized Maximum Entropy Estimation in Measurement Error Models

Generalized maximum entropy approach is proposed by Golan et al. [6] to overcome the problem of collinearity and ill-conditioning among the explanatory variables in ordinary linear models. In this paper we extend this approach to the functional measurement error models. We derive generalized maximum entropy and generalized cross entropy estimate of parameters in this context. We illustrate our idea via an example from real data. Finally, based on this data set we perform a parametric bootstrap simulation study to compare entropy estimators with ridge and maximum likelihood estimators.

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.

Estimation and Testing in Elliptical Functional Measurement Error Models

The purpose of this article is to investigate estimation and hypothesis testing by maximum likelihood and method of moments in functional models within the class of elliptical symmetric distributions. The main results encompass consistency and asymptotic normality of the method of moments estimators. Also, the asymptotic covariance matrix of the maximum likelihood estimator is derived, extending some existing results in elliptical distributions. A measure of asymptotic relative efficiency is reported. Wald-type statistics are considered and numerical results obtained by Monte Carlo simulation to investigate the performance of estimators and tests are provided for Student-t and contaminated normal distributions. An application to a real dataset is also included.

Variable Selection in Measurement Error Models.

Measurement error data or errors-in-variable data are often collected in many studies. Natural criterion functions are often unavailable for general functional measurement error models due to the lack of information on the distribution of the unobservable covariates. Typically, the parameter estimation is via solving estimating equations. In addition, the construction of such estimating equations routinely requires solving integral equations, hence the computation is often much more intensive compared with ordinary regression models. Because of these difficulties, traditional best subset variable selection procedures are not applicable, and in the measurement error model context, variable selection remains an unsolved issue. In this paper, we develop a framework for variable selection in measurement error models via penalized estimating equations. We first propose a class of selection procedures for general parametric measurement error models and for general semiparametric measurement error models, and study the asymptotic properties of the proposed procedures. Then, under certain regularity conditions and with a properly chosen regularization parameter, we demonstrate that the proposed procedure performs as well as an oracle procedure. We assess the finite sample performance via Monte Carlo simulation studies and illustrate the proposed methodology through the empirical analysis of a familiar data set.

Influential Observations in the Functional Measurement Error Model

In this work we propose Bayesian measures to quantify the influence of observations on the structural parameters of the simple measurement error model (MEM). Different influence measures, like those based on q-divergence between posterior distributions and Bayes risk, are studied to evaluate the influence. A strategy based on the perturbation function and MCMC samples is used to compute these measures. The samples from the posterior distributions are obtained by using the Metropolis–Hastings algorithm and assuming specific proper prior distributions. The results are illustrated with an application to a real example modeled with MEM in the literature.

Semiparametric estimators of functional measurement error models with unknown error

SummaryWe consider functional measurement error models where the measurement error distribution is estimated non-parametrically. We derive a locally efficient semiparametric estimator but propose not to implement it owing to its numerical complexity. Instead, a plug-in estimator is proposed, where the measurement error distribution is estimated through non-parametric kernel methods based on multiple measurements. The root n consistency and asymptotic normality of the plug-in estimator are derived. Despite the theoretical inefficiency of the plug-in estimator, simulations demonstrate its near optimal performance. Computational advantages relative to the theoretically efficient estimator make the plug-in estimator practically appealing. Application of the estimator is illustrated by using the Framingham data example.

Local influence assessment in heteroscedastic measurement error models

Functional heteroscedastic measurement error models are investigated aiming to assess the effects of perturbations of data on some inferential procedures. This goal is accomplished by resorting to methods of local influence. The techniques provide to the practitioner a valuable tool that enables to identify potential influential elements and to quantify the effects of perturbations in these elements on results of interest. An illustrative example with a real data set is also reported.

Empirical Bayes estimation in finite population sampling under functional measurement error models

The paper considers simultaneous estimation of finite population means for several strata. A model-based approach is taken, where the covariates in the super-population model are subject to measurement errors. Empirical Bayes (EB) estimators of the strata means are developed and an asymptotic expression for the MSE of the EB estimators is provided. It is shown that the proposed EB estimators are “first order optimal” in the sense of Robbins [1956. An empirical Bayes approach to statistics. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, University of California Press, Berkeley, pp. 157–164], while the regular EB estimators which ignore the measurement error are not.

Empirical process based on the recursive residuals in functional measurement error models

Recursive residuals are a linear transformation of ordinary residuals. They have frequently been suggested for testing model fit and model assumptions in linear regression. In this paper, we generalize the theory of the empirical process based on the residuals in the measurement error models to the recursive residuals in the measurement error models. We prove that recursive residuals in these models are asymptotically independent and identically distributed and show that, in this case, the weak convergence properties of the standardized residuals hold for the studentized recursive residuals. Furthermore, we look at some tests for goodness-of-fit based on the weak convergence of the empirical distribution of the recursive residuals. We justify the use of goodness-of-fit tests based on recursive residuals by carrying out a parametric bootstrap simulation study.

Locally efficient semiparametric estimators for functional measurement error models

SUMMARY A class of semiparametric estimators are proposed in the general setting of functional measurement error models. The estimators follow from estimating equations that are based on the semiparametric efficient score derived under a possibly incorrect distri butional assumption for the unobserved 'measured with error' covariates. It is shown that

INFLUENCE FUNCTIONS IN FUNCTIONAL MEASUREMENT ERROR MODELS WITH REPLICATED DATA

We consider the construction and properties of influence functions in the context of functional measurement error models with replicated data. In these models estimates of the parameters can be affected both by the individual observations and the means of replicated observations. We show that influence function of the means of replicates on the estimate of regression coefficients can be only derived under the assumption that the variances of the errors are known, while one for the individual observations can be only derived simultaneously with their influence function on the estimators of the variances of the errors.