Linear Measurement Error Models Research Articles

Influence measures and outliers detection in linear mixed measurement error models with Ridge estimation

In this article, we have concentrated on the influence diagnostics based on the mean shift outlier models (MSOM) and case deletion methods in the linear mixed measurement error models with Ridge estimation using the corrected likelihood of Nakamura (1990). We have derived the corrected score test statistic for outliers detection according to MSOM. Then, several case deletion measures are constructed as a tool for influence diagnostics. A parametric bootstrap procedure is used to obtain empirical distributions of the test statistics, and a simulation study has been used to show the performance of the proposed score test statistic and cook distance. Finally, a numerical example is given to illustrate the theoretical results.

Inference in high dimensional linear measurement error models

For a high dimensional linear model with a finite number of covariates measured with errors, we study statistical inference on the parameters associated with the error-prone covariates, and propose a new corrected decorrelated score test and a corresponding score type estimator. This work was motivated by a real data example, where both low dimensional phenotypic variables and high dimensional genotypic variables, single nucleotide polymorphisms (SNPs), are available. One of the phenotypic variables is of clinical interest but measured with error. As is standard in the literature, the high dimensional SNPs are assumed to be measured accurately.We show that the limiting distribution of our corrected decorrelated score test statistic is standard normal under the null hypothesis and retains power under the local alternatives around zero. We also establish the asymptotic normality of the newly proposed estimator, and hence asymptotic confidence intervals can be constructed. The finite-sample performance of the proposed inference procedure is examined through simulation studies. We further illustrate the proposed procedure via an empirical analysis of the real data example mentioned above.

The Mixed Liu Estimator in Stochastic Restricted Linear Measurement Error Model

Ghapani and Babdi [1] proposed a mixed Liu estimator in linear measurement error model with stochastic linear restrictions. In this article, we propose an alternative mixed Liu estimator in the linear measurement error model with stochastic linear restrictions. The performance of the new mixed Liu estimator over the mixed estimator, Liu estimator, and mixed Liu estimator proposed by Ghapani and Babdi [1] are discussed in the sense of mean squared error matrix. Finally, a simulation study is given to show the performance of these estimators.

Assessment of local influence in spatial elliptical linear measurement error models

In this paper, we consider estimation and inference procedures in spatial linear models when some of the covariates are measured with errors. It is assumed that the additive error distributed according to the law belonging to the class of elliptically contoured distributions. The development of the corrected score function with the family of elliptical distributions is the basis for derivation of the estimators. For sensitivity analysis, the local influence approach is used for assessing influence of small perturbations on the parameter estimates. A simulation study is presented illustrating the good performance of the proposed approach, including the robustness property for the heavier tail models. We further illustrate the proposed procedures by an application in a real dataset.

Online updating method to correct for measurement error in big data streams

When huge amounts of data arrive in streams, online updating is an important method to alleviate both computational and data storage issues. The scope of previous research for online updating is extended in the context of the classical linear measurement error model. In the case where some covariates are unknowingly measured with error at the beginning of the stream, but then are measured without error after a particular point along the data stream, the updated estimators ignoring the measurement error are biased for the true parameters. Once the covariates measured without error are first observed, a method to correct the bias of the estimators, as well as to correct the biases in their variance estimator, is proposed; after correction, the traditional online updating method can then proceed as usual. Further, asymptotic distributions for the corrected and updated estimators are established. Simulation studies and a real data analysis with an airline on-time dataset are provided to illustrate the performance of the proposed method.

Logarithmic calibration for partial linear models with multiplicative distortion measurement errors

ABSTRACTIn this paper, we propose a new identifiability condition by using the logarithmic calibration for the multiplicative distortion partial linear measurement errors models, when neither the response variable nor the covariates in the parametric part can be directly observed. We propose a logarithmic calibration estimation procedure for the unobserved variables. Then, a profile least squares estimator is proposed, associated with its asymptotic results and confidence intervals construction. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties for the estimator and test statistic are also established. We employ the smoothly clipped absolute deviation penalty to select relevant variables. Simulation studies demonstrate the performance of the proposed procedure and a real example is analysed to illustrate its practical usage.

Improved estimation in elliptical linear mixed measurement error models

In this paper, we propose a set of improved estimators for the fixed effect parameters in the linear mixed models when the covariates are measured with additive errors and prior information for the parameters is available. When it is suspected that the parameter vector may be the null-vector with some degree of uncertainty, we define improved estimators which including the preliminary test estimator, the Stein-type estimator and the postive-rule Stein-type estimator. It is assumed that the measurement error distributed according to the law belonging to the class of elliptically contoured distributions. The asymptotic properties of resulting estimators such as the asymptotic distributional quadratic biases and the asymptotic distributional quadratic risks are examined. A simulation study is also performed to illustrate the finite sample performance of the proposed procedures and finally Boston housing data set is analysed.

Estimation and Inference in Semi-Functional Partially Linear Measurement Error Models

This article studies the estimation and statistical inference problems of semi-functional partially linear regression models when the covariates in the linear part are measured with additive error. To obtain the estimation of the parametric component, a corrected profile least-squares based estimation procedure is developed. Asymptotic properties of the proposed estimators are established under some mild assumptions. To test hypothesis on the parametric part, the authors propose a novel test statistic based on the difference between the corrected residual sums of squares under the null and alternative hypotheses, and show that its limiting distribution is a weighted sum of independent standard χ12 . Finally, the authors illustrate the finite sample performance of the methods with some simulation studies and a real data application.

Ridge estimation in linear mixed measurement error models with stochastic linear mixed restrictions

This article is concentrated on the problem of multicollinearity in linear mixed models (LMMs) with measurement error in the fixed effects variables. After introducing a ridge estimator (RE) in these models, we propose a new estimator called the stochastic restricted ridge estimator (SRRE) by combining the ridge estimator (RE) and the stochastic restricted estimator (SRE). Moreover, asymptotic properties of these estimators will be derived and the necessary and sufficient conditions for the superiority of the SRRE over the RE and SRE are obtained using the mean squared error matrix (MSEM). Finally, the theoretical findings of the proposed estimators are also evaluated with a Monte Carlo simulation study and a numerical example.

A Note on Testing Serial Correlation in Partially Linear Additive Measurement Error Models

The paper considers the problem of testing error serial correlation of partially linear additive measurement error model. We propose a test statistic and show that it converges to the standard chi-square distribution under the null hypothesis. Finally, a simulation study is conducted to illustrate the performance of the test approach.

Stochastic restricted Liu estimator in linear mixed measurement error models

This paper is concerned with the Liu estimator, stochastic restricted estimator and stochastic restricted Liu estimator of fixed and random effects in the linear mixed measurement error models. We compare the proposed estimators under the criterion of mean squared error matrix (MSEM). Furthermore, the selection of the Liu biasing parameter is discussed. A real data analysis is provided to illustrate the theoretical results and a simulation study is conducted to characterize the performance of estimators in the linear mixed measurement error model.

Using Liu estimator for detection of influential observations in linear measurement error models

In this paper, we introduce Liu estimator for the vector of parameters in linear measurement error models and discuss its asymptotic properties. Based on the Liu estimator, diagnostic measures are developed to identify influential observations. Additionally, the analogs of Cook’s distance and likelihood distance are proposed to determine influential observations using case deletion approach. A parametric bootstrap procedure is used to obtain empirical distributions of the test statistics. Finally, the performance of the influence measures have been illustrated through simulation study and analyzing a real data set.

Conditional absolute mean calibration for partial linear multiplicative distortion measurement errors models

In this paper we consider partial linear regression models when all the variables are measured with multiplicative distortion measurement errors. To eliminate the effect caused by the distortion, we propose the conditional absolute mean calibration, which avoids to use the nonzero expectation conditions imposed on the variables. With these calibrated variables, a profile least squares estimator is obtained, associated with its normal approximation based and empirical likelihood based confidence intervals. For the hypothesis testing on parameters, a restricted estimator under the null hypothesis and a test statistic are proposed. A smoothly clipped absolute deviation penalty is employed to select the relevant variables. The resulting penalized estimators are shown to be asymptotically normal and have the oracle property. Lastly, a score-type test statistic is then proposed for checking the validity of partial linear models. We derive asymptotic distribution of the proposed test statistic. The quadratic form of the scaled test statistic has an asymptotic chi-squared distribution under the null hypothesis and follows a noncentral chi-squared distribution under local alternatives, which converge to the null hypothesis at a parametric rate. Simulation studies demonstrate the performance of our proposed procedure and a real example is analyzed as illustrate its practical usage.

Estimation and variable selection for partial linear single-index distortion measurement errors models

This paper considers partial linear single-index regression models when all the variables are measured with multiplicative distortion measurement errors. To eliminate the effect caused by the distortion, we propose the conditional absolute mean calibration. This method avoids to use the nonzero expectation conditions imposed on the variables in the literature. Using the calibrated variables, a profile least squares estimator is obtained. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. A smoothly clipped absolute deviation penalty is employed to select the relevant variables. The resulting penalized estimators are shown to be asymptotically normal and have the oracle property. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its practical usage.

MALMEM: model averaging in linear measurement error models.

We develop model averaging estimation in the linear regression model where some covariates are subject to measurement error. The absence of the true covariates in this framework makes the calculation of the standard residual-based loss function impossible. We take advantage of the explicit form of the parameter estimators and construct a weight choice criterion. It is asymptotically equivalent to the unknown model average estimator minimizing the loss function. When the true model is not included in the set of candidate models, the method achieves optimality in terms of minimizing the relative loss, whereas, when the true model is included, the method estimates the model parameter with root n rate. Simulation results in comparison with existing Bayesian information criterion and Akaike information criterion model selection and model averaging methods strongly favour our model averaging method. The method is applied to a study on health.

Improved statistical inference on semiparametric varying-coefficient partially linear measurement error model

ABSTRACT In this paper, we consider the estimation and goodness-of-fit test of a semiparametric varying-coefficient partially linear (SVCPL) model when both responses and part of covariates are measured with error. It is assumed that the true variables are measurable functions of some auxiliary variables. The often-used assumptions on the measurement error, such as a known error variance, a known distribution of the error variable, a validation sample or a repeated data set, are not required. The asymptotic properties of the proposed estimators and testing statistic are investigated. We show that the application of the measurement error structures can improve the efficiency of estimating and testing methods. The performances of the estimating and testing methods are illustrated by simulation studies and an application to a real data set.

Oracally efficient spline-backfitted kernel smoothing of additive partial linear measurement error model

We consider statistical inference for additive partial linear models when the linear covariate is measured with error. A bias-corrected spline-backfitted kernel smoothing method is proposed. Under mild assumptions, the proposed component function and parameter estimator are oracally efficient and fast to compute. The nonparametric function estimator’s pointwise distribution is asymptotically equivalent to an function estimator in partial linear model. Finite-sample performance of the proposed estimators is assessed by simulation experiments. The proposed methods are applied to Boston house data set.

There is no impact of exposure measurement error on latency estimation in linear models.

Identification of the latency period for the effect of a time-varying exposure is key when assessing many environmental, nutritional, and behavioral risk factors. A pre-specified exposure metric involving an unknown latency parameter is often used in the statistical model for the exposure-disease relationship. Likelihood-based methods have been developed to estimate this latency parameter for generalized linear models but do not exist for scenarios where the exposure is measured with error, as is usually the case. Here, we explore the performance of naive estimators for both the latency parameter and the regression coefficients, which ignore exposure measurement error, assuming a linear measurement error model. We prove that, in many scenarios under this general measurement error setting, the least squares estimator for the latency parameter remains consistent, while the regression coefficient estimates are inconsistent as has previously been found in standard measurement error models where the primary disease model does not involve a latency parameter. Conditions under which this result holds are generalized to a wide class of covariance structures and mean functions. The findings are illustrated in a study of body mass index in relation to physical activity in the Health Professionals Follow-Up Study.

Statistical inference for linear regression models with additive distortion measurement errors

We consider estimations and hypothesis test for linear regression measurement error models when the response variable and covariates are measured with additive distortion measurement errors, which are unknown functions of a commonly observable confounding variable. In the parameter estimation and testing part, we first propose a residual-based least squares estimator under unrestricted and restricted conditions. Then, to test a hypothesis on the parametric components, we propose a test statistic based on the normalized difference between residual sums of squares under the null and alternative hypotheses. We establish asymptotic properties for the estimators and test statistics. Further, we employ the smoothly clipped absolute deviation penalty to select relevant variables. The resulting penalized estimators are shown to be asymptotically normal and have the oracle property. In the model checking part, we suggest two test statistics for checking the validity of linear regression models. One is a score-type test statistic and the other is a model- adaptive test statistic. The quadratic form of the scaled test statistic is asymptotically chi-squared distributed under the null hypothesis and follows a noncentral chi-squared distribution under local alternatives that converge to the null hypothesis. We also conduct simulation studies to demonstrate the performance of the proposed procedure and analyze a real example for illustration.

Elliptical difference based ridge and Liu type estimators in partial linear measurement error models

In this article the difference based ridge and Liu type estimators are defined in partial linear models when the covariates are measured with additive errors. It is assumed that the additive error distributed according to the law belonging to the class of elliptically contoured distributions. The development of the corrected score function with the family of elliptical distributions is the basis for derivation of the estimators. The asymptotic normality of the estimates are established. Also, the conditions for the superiority of the proposed estimator over its counterpart, for selecting the biasing parameters are obtained. A simulation study is also performed.