Multiplicative Measurement Errors Research Articles

Fitting data to a multiple structural measurement errors model

This paper proposes two new estimation methods to fit a multiple structural measurement error model when all variables are subject to errors. The new estimation methods were extensions of the Wald estimation method, one is the weighted grouping method, and the other is the iterative method. A Monte Carlo experiment is performed to investigate the performance of the new estimators compared with the classical estimation methods; the Maximum Likelihood Estimator and Method of Moment, in terms of root mean square error and its bias. The simulation outcomes demonstrated that the suggested estimators are more effective than conventional estimators. In addition, real data analysis is discussed to examine the relationship between the national gross domestic product, unemployment rate, and human development index after applying the two proposed estimation methods.

Testing distribution for multiplicative distortion measurement errors

. In this article, we study a goodness of fit test for a multiplicative distortion model under a uniformly distributed but unobserved random variable. The unobservable variable is distorted in a multiplicative fashion by an observed confounding variable. The proposed k-th power test statistic is based on logarithmic transformed observations and a correlation coefficient-based estimator without distortion measurement errors. The proper choice of k is discussed through the empirical coverage probabilities. The asymptotic null distribution of the test statistics are obtained with known asymptotic variances. Next, we proposed the conditional mean calibrated test statistic when a variable is distorted in a multiplicative fashion. We conduct Monte Carlo simulation experiments to examine the performance of the proposed test statistics.

Jackknife empirical likelihood for the correlation coefficient with multiplicative distortion measurement errors

In this paper, we consider the estimation problem of a correlation coefficient between two unobserved variables of interest that are distorted in a multiplicative way by some unobserved confounding variable. We investigate the direct plug-in estimator of the correlation coefficient. We propose using jackknife empirical likelihood (JEL) and its variations to construct confidence intervals for the correlation coefficient based on the estimator. The proposed JEL statistic is shown to be asymptotically a standard chi-squared distribution. We compare our methods to the previous empirical likelihood (EL) techniques of Zhang et al. (2014, ‘A Revisit to Correlation Analysis for Distortion Measurement Error Data’, Journal of Multivariate Analysis, 124, 116–129) and show the JEL possesses better small sample properties. Simulation studies are conducted to examine the performance of the proposed estimator, and we also use our proposed methods to analyse the Boston housing data for illustration.

Coverage Control for Mobile Sensor Networks with Multiplicative Measurement Errors and Limited Interaction Ranges

Coverage Control for Mobile Sensor Networks with Multiplicative Measurement Errors and Limited Interaction Ranges

Covariance ratio under multiplicative distortion measurement errors

We propose a covariance ratio measure for symmetry or asymmetry of a probability density function. This measure is constructed by the ratio of the covariance connected with the density function and the distribution function. We first propose a non parametric moment-based estimator of the covariance ratio measure and study its asymptotic results. Next, we consider statistical inference of the covariance ratio measure by using the empirical likelihood method. The empirical likelihood statistic is shown to be asymptotically a standard chi-squared distribution. Last, we study the covariance ratio measure when the random variable is unobserved under the multiplicative distortion measurement errors setting. The density function and the distribution function of the unobserved variable are estimated by using four calibrated variables. Appealing to these estimators, the calibrated covariance ratio measures are proposed and further shown to be asymptotically efficient as if there are no multiplicative distortion effects. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators and test procedures. These methods are applied to analyze two real datasets for illustration.

Consistency of the fixed effects Poisson estimator with multiplicative measurement error and unbalanced panels

We study the asymptotic properties of the popular fixed effects Poisson (FEP) estimator in two scenarios important for empirical research. First, we allow for multiplicative measurement error and characterize when the FEP estimator consistently estimates the parameters. Second, we allow for unbalanced panels and characterize when the estimator is consistent under the complete cases. We propose simple tests to determine whether selection is systematically related to unobserved time-varying shocks, which can cause the FEP estimator to be inconsistent. A simulation shows that the predictions of the theoretical findings hold in reasonable sample sizes.

Testing symmetry of model errors for non linear multiplicative distortion measurement error models

To study the symmetry and asymmetry of the model error under multiplicative distortion measurement errors setting, we propose a correlation coefficient-based measure between the distribution function and the root of density function. The unknown distribution function and density function are estimated from four kinds of residuals: the conditional mean calibration-based residuals, the conditional absolute mean calibration-based residuals, the conditional variance calibration-based residuals, and the conditional absolute logarithmic calibration-based residuals. We study the asymptotic results of the estimators of correlation coefficient-based measure under four calibrations. Next, we consider statistical inference of the correlation coefficient-based measure by using the empirical likelihood method. The empirical likelihood statistics are shown to be an asymptotically standard chi-squared distribution. Simulation studies demonstrate the performance of the proposed estimators and test statistics. A real example is analyzed to illustrate its practical usage.

Parametric hypothesis tests for exponentiality under multiplicative distortion measurement errors data

In this paper, we proposed a parametric hypothesis test of the multiplicative distortion model under the exponentially distributed but unobserved random variable. The unobservable variable is distorted in a multiplicative fashion by an observed confounding variable. Firstly, some new test statistics are proposed to checking the exponential distribution assumption without distortion effects. Next, we proposed several test statistics when the variable is distorted in the multiplicative fashion. For the latter, the proposed test statistics automatically eliminate the distortion effects involved in the unobserved variable. The proposed test statistics with or without distortions are all asymptotical free, and the asymptotic null distribution of the test statistics are obtained with known asymptotic variances. We conduct Monte Carlo simulation experiments to examine the performance of the proposed test statistics. These methods are applied to analyze four real datasets for illustrations.

Linear regression models with multiplicative distortions under new identifiability conditions

This paper considers linear regression models when neither the response variable nor the covariates can be directly observed, but are measured with multiplicative distortion measurement errors. We propose new identifiability conditions for the distortion functions via the varying coefficient models, then moment‐based estimators of parameters in the model are proposed by using the estimated varying coefficient functions. This method does not require the independence condition between the confounding variables and the unobserved response and variables. We establish the connections among the varying coefficient based estimators, the conditional mean calibration and the conditional absolute mean calibration. We study the asymptotic results of these proposed estimators, and discuss their asymptotic efficiencies. Lastly, we make some comparisons among the proposed estimators through the simulation. These methods are applied to analyze a real dataset for an illustration.

A parametric hypothesis test for a global power law and local nonparametric trend model with multiplicative distortion measurement errors

We consider a parametric hypothesis test of the multiplicative distortion model with both a parametric global trend and a nonparametric local trend. In the model, neither the response variable nor the covariates can be directly observed, but are measured with multiplicative distortion measurement errors. Four calibration procedures with several estimation methods are used to construct the test statistics, namely, conditional mean calibration based test statistic, conditional absolute mean calibration based test statistic, conditional variance calibration based test statistic, and conditional absolute logarithmic calibration based test statistic. Asymptotic properties for these test statistics are established under the null hypothesis and alternative hypothesis. Monte Carlo simulation experiments are carried out to examine the performance of the proposed test statistics. A real example is analyzed to illustrate their practical usages.

Gaussian graphical model estimation with measurement error

It is well known that regression methods designed for clean data will lead to erroneous results if directly applied to corrupted data. Despite the recent methodological and algorithmic advances in Gaussian graphical model estimation, how to achieve efficient and scalable estimation under contaminated covariates is unclear. Here a new methodology called convex conditioned innovative scalable efficient estimation (COCOISEE) for Gaussian graphical model under both additive and multiplicative measurement errors is developed. It combines the strengths of the innovative scalable efficient estimation in Gaussian graphical model and the nearest positive semi-definite matrix projection, thus enjoying stepwise convexity and scalability. Comprehensive theoretical guarantees are provided and the effectiveness of the proposed methodology is demonstrated through numerical studies.

Optimal Pattern Synthesis for Polarimetric Phased Array Antenna Using Representative Array Patterns

To reduce pattern measurements and synthesis computations for an optimal array pattern, a set of full array patterns, named as representative array patterns (RAPs), is proposed, the number of which is smaller than that of the active element patterns (AEPs). Each RAP corresponds to one beam named as the representative beam. By optimizing the directions of these representative beams, the measured RAPs can be a good substitute for all AEPs. Compared with using the AEPs, the measurement cost and the computation time of the proposed method are proportionally cut. Considering the results of multiple evaluated directions, the optimal arrangement of representative beams is analyzed. Relative to using the AEPs, a highly consistent performance is achieved, of which the correlation coefficient of copolarization patterns can exceed 98%. For the simulated 10 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times 10$ </tex-math></inline-formula> array antenna, since merely 67 RAPs are utilized, a 33% reduction in pattern measurements and synthesis computations is obtained. This relative reduction ratio reaches 39% for the 16 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times 16$ </tex-math></inline-formula> array antenna and 50% for the 32 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times 32$ </tex-math></inline-formula> array antenna. The stability of the proposed method under multiplicative and additive pattern measurement errors is presented. Moreover, the proposed method is verified on a practical C-band 4 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times 4$ </tex-math></inline-formula> dual-polarized microstrip patch array antenna.

Sparse Hanson–Wright inequality for a bilinear form of sub‐Gaussian variables

In this paper, we derive a new version of Hanson–Wright inequality for a sparse bilinear form of sub‐Gaussian variables. Our results are a generalization of previous deviation inequalities that consider either sparse quadratic forms or dense bilinear forms. We apply the new concentration inequality to testing the cross‐covariance matrix when data are subject to missing. Using our results, we can find a threshold value of correlations that controls the family‐wise error rate. Furthermore, we discuss the multiplicative measurement error case for the bilinear form with a boundedness condition.

Nonparametric multiplicative distortion measurement errors models with bias reduction

In this paper we propose a new nonparametric regression technique under multiplicative distortion measurement errors settings. The unobservable variables are both distorted in a multiplicative fashion by an observed confounding variable. A bias reduction is proposed by choosing a function that is the projection of the unknown regression function onto the parametric family in a certain metric. We find that this new technique leads to substantial improvement in the performance of regression estimators in comparison with the direct one-step estimation, irrespective of the choice of a parametric model. We obtain asymptotic normality results for the estimated nonparametric kernel smoothers, and further discuss their estimation efficiency. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators.

Anisotropic spectral cut-off estimation under multiplicative measurement errors

We study the non-parametric estimation of an unknown density f with support on R+d based on an i.i.d. sample with multiplicative measurement errors. The proposed fully-data driven procedure is based on the estimation of the Mellin transform of the density f and a regularization of the inverse of Mellin transform by a spectral cut-off. The bias–variance tradeoff for estimating f is optimized with a data-driven anisotropic choice of the cutoff parameter. In order to discuss the bias term, we consider the Mellin–Sobolev spaces which define the regularity of the unknown density f through the decay of its Mellin transform. Additionally, we show minimax-optimality over Mellin–Sobolev spaces of the spectral cut-off density estimator.

Multiplicative deconvolution in survival analysis under dependency

We study the non-parametric estimation of an unknown survival function S with support on R + based on a sample with multiplicative measurement errors. The proposed fully data-driven procedure is based on the estimation of the Mellin transform of the survival function and a regularization of the inverse of the Mellin transform by a spectral cut-off. The upcoming bias-variance trade-off is handled by a data-driven choice of the cut-off parameter. In order to discuss the bias term, we consider the Mellin–Sobolev spaces which characterize the regularity of the unknown survival function S through the decay of its Mellin transform. For the analysis of the variance term, we consider the independent and identically distributed case and incorporate dependent observations in form of Bernoulli shift processes and β-mixing sequences. Additionally, we show minimax optimality over Mellin–Sobolev spaces of the spectral cut-off estimator.

Estimation of correlation coefficient under a linear multiplicative distortion measurement errors model

This paper studies the estimation of correlation coefficient between unobserved variables of interest. These unobservable variables are distorted in a new multiplicative fashion by an observed confounding variable, namely, a parametric distortion measurement errors model. The least squares estimation and weighted least squares estimation are used to estimate parameters in the multiplicative distortion functions. An interesting finding is that the calibrated variables obtained by using the weighted least squares estimators perform asymptotic equivalent to the calibrated variables obtained from the nonparametric kernel smoothing estimators in the literature. We then used the calibrated variables to estimate the correlation coefficient, and show that the proposed estimators of correlation coefficient are all asymptotically efficient. Moreover, we suggest an asymptotic normal approximation and an empirical likelihood-based statistic to construct the confidence intervals. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators. These methods are applied to analyze two real datasets for an illustration.

Nonlinear multiplicative distortion regression models with second-order estimation

This paper considers nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with multiplicative distortion measurement errors. We proposed the second order least squares estimator by including in the criterion function the distance of both the response variable to its conditional mean and the squared response variable to its second conditional moment given the predictor variables. The asymptotic results of the second order nonlinear least squares estimators for the regression parameter are derived. As a by-product, we construct confidence intervals by estimating the asymptotic covariance matrices. We also consider to use three existing calibration procedures to calibrate the unknown response and covariates, namely, the conditional absolute mean calibration, the conditional variance calibration and the logarithmic calibration. The simulation studies shows that these second order nonlinear least squares estimators with three calibration procedures are all superior than the ordinary nonlinear least squares estimators.

Average derivation estimation with multiplicative distortion measurement errors

We consider the average derivation estimation when neither the response variable nor the covariates can be directly observed, but are measured with multiplicative distortion measurement errors. Three calibration procedures with several estimation methods are used to estimate the average derivation, namely, conditional mean calibration, conditional absolute mean calibration and conditional variance calibration. The asymptotic properties for the estimators are established. Monte Carlo simulation experiments are conducted to examine the performance of the proposed estimators.

Modal linear regression models with multiplicative distortion measurement errors

AbstractWe consider modal linear regression models when neither the response variable nor the covariates can be directly observed, but are measured with multiplicative distortion measurement errors. Four calibration procedures are used to estimate parameters in the modal linear regression models, namely, conditional mean calibration, conditional absolute mean calibration, conditional variance calibration, and conditional absolute logarithmic calibration. The asymptotic properties for the estimators based on four calibration procedures are established. Monte Carlo simulation experiments are conducted to examine the performance of the proposed estimators. The proposed estimators are applied to analyze a forest fires dataset for an illustration.