Additive distortion measurement errors regression models with exponential calibration

We consider partial linear regression models when all the variables are measured with additive distortion measurement errors. To eliminate the effect caused by the distortion, we propose the conditional mean calibration to obtain calibrated variables. A profile least squares estimator for the parameter 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. Simulation studies demonstrate the performance of our proposed procedure and a real example is analyzed as illustrate its practical usage.

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.

Partial differential equation (PDE) models are often used to study biological phenomena involving movement-birth–death processes, including ecological population dynamics and the invasion of populations of biological cells. Count data, by definition, is non-negative, and count data relating to biological populations is often bounded above by some carrying capacity that arises through biological competition for space or nutrients. Parameter estimation, parameter identifiability, and making model predictions usually involves working with a measurement error model that explicitly relating experimental measurements with the solution of a mathematical model. In many biological applications, a typical approach is to assume the data are normally distributed about the solution of the mathematical model. Despite the widespread use of the standard additive Gaussian measurement error model, the assumptions inherent in this approach are rarely explicitly considered or compared with other options. Here, we interpret scratch assay data, involving migration, proliferation and delays in a population of cancer cells using a reaction–diffusion PDE model. We consider relating experimental measurements to the PDE solution using a standard additive Gaussian measurement error model alongside a comparison to a more biologically realistic binomial measurement error model. While estimates of model parameters are relatively insensitive to the choice of measurement error model, model predictions for data realisations are very sensitive. The standard additive Gaussian measurement error model leads to biologically inconsistent predictions, such as negative counts and counts that exceed the carrying capacity across a relatively large spatial region within the experiment. Furthermore, the standard additive Gaussian measurement error model requires estimating an additional parameter compared to the binomial measurement error model. In contrast, the binomial measurement error model leads to biologically plausible predictions and is simpler to implement. We provide open source Julia software on GitHub to replicate all calculations in this work, and we explain how to generalise our approach to deal with coupled PDE models with several dependent variables through a multinomial measurement error model, as well as pointing out other potential generalisations by linking our work with established practices in the field of generalised linear models.

This paper studies how to estimate and test the symmetry of a continuous variable under the additive distortion measurement errors setting. The unobservable variable is distorted in a additive fashion by an observed confounding variable. In this paper, a direct plug-in estimation procedure and two-step direct plug-in estimation procedure for correlation coefficient are proposed to measure the symmetry or asymmetry of a continuous variable, and empirical likelihood based confidence intervals are constructed to test the symmetry of the unobserved variable. The asymptotic properties of the proposed estimators and test statistics are investigated. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators and test statistics. These methods are applied to analyze a real dataset for an illustration.

In this article, we propose a new identifiability condition by using the logarithmic calibration for the distortion measurement error models, where neither the response variable nor the covariates can be directly observed but are measured with multiplicative measurement errors. Under the logarithmic calibration, the direct‐plug‐in estimators of parameters and empirical likelihood based confidence intervals are proposed, and we studied the asymptotic properties of the proposed estimators. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties for the restricted estimator and test statistic are established. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its practical usage.

This paper considers the estimation for a partial index additive regression model, when the response variable and covariates in the index part are observed with additive distortion measurement errors. For the index parameter, the dimension-reduction based estimators with or without additive distortion measurement errors are proposed. This new estimation method is further adopted to the partial linear models for parameter estimation. We study the asymptotic properties of the proposed estimators. Simulation studies are conducted to compare the proposed estimation methods.

This paper studies the estimation of correlation coefficient between unobserved variables of interest. These unobservable variables are distorted in an additive fashion by an observed confounding variable. We propose a new identifiability condition by using the exponential calibration to obtain calibrated variables and propose a direct‐plug‐in estimator for the correlation coefficient. We show that the direct‐plug‐in estimator is asymptotically efficient. Next, we suggest an asymptotic normal approximation and an empirical likelihood‐based statistic to construct the confidence intervals. Last, we propose several test statistics for testing whether the true correlation coefficient is zero or not. The asymptotic properties of the proposed test statistics are examined. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators and test statistics. These methods are applied to analyze a temperature forecast data set for an illustration.

Estimation and hypothesis test on partial linear models with additive distortion measurement errors

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

We study the goodness-of-fit tests for checking the normality of the model errors under the additive distortion measurement error settings. Neither the response variable nor the covariates can be directly observed but are distorted in additive fashions by an observed confounding variable. The proposed test statistics is based on logarithmic transformed variables with residuals and a particular choice of the kth power covariance-based estimator. The proposed test statistics has three advantages. Firstly, the asymptotic null distribution of the test statistics are obtained with known asymptotic variance. Secondly, the test statistic tests are irrelevant to the model. Thirdly, the proposed test statistics automatically eliminate the additive distortion effects involved in the response and covariates. The simulation studies show the proposed test statistics can be used to check normality when the sample size is very large. A real example is analysed to illustrate its practical usage.

This paper studies how to estimate and test the symmetry of a continuous variable under the additive distortion measurement errors setting. The unobservable variable is distorted in a additive fashion by an observed confounding variable. Firstly, a direct plug-in estimation procedure is proposed to calibrate the unobserved variable. Next, we propose four test statistics for testing whether the unobserved variable is symmetric or not. The asymptotic properties of the proposed estimators and test statistics are investigated. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators and test statistics. These methods are applied to analyze a real dataset for an illustration.

Empirical likelihood inference for the mean residual life under random censorship

Measurement data in the field of modern geodesy contains not only additive errors but also multiplicative errors related to signal strength. The existing models for dealing with mixed additive and multiplicative errors are mainly based on the linear form of unknown parameters and observations, and there are few studies on the nonlinear form of the two. In the parameter estimation method of the nonlinear mixed additive and multiplicative errors model, the initial value of the Gauss–Newton parameter estimation method is selected by previous experience. The initial value determined by this method deviates far from the true value due to a lack of experience, which will lead to inaccurate parameter estimation results. In order to solve this problem, based on the least squares principle and the introduction of the damping factor, this paper deduces the damping least squares parameter solution formula for the parameter estimation of the nonlinear mixed additive and multiplicative errors model. The superiority of the damping least squares algorithm is reflected in the adjustment of the damping factor, taking into account the advantages of the Gauss–Newton method and the steepest descent method, and some weighted selection is obtained in the two algorithms. The calculation and comparative analysis of the simulated cases show that the damped least squares method is more suitable for handling geodetic data with this nonlinear mixed additive and multiplicative errors model when the initial value deviates far from the true value.

This paper considers partial linear regression models when neither the response variable nor the covariates can be directly observed, but are instead measured with both multiplicative and additive distortion measurement errors. We propose conditional variance estimation methods to calibrate the unobserved variables. A profile least-squares estimator associated with the asymptotic results and confidence intervals is then proposed. To do hypothesis testing of the parameters, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties of the estimator and the test statistic are also established. 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. Estimation, hypothesis testing, and variable selection are discussed under the scenario of multiplicative distortion alone. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its applicability.

This paper considers nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with both multiplicative and additive distortion measurement errors. We propose conditional variance and conditional mean calibration estimation methods for the unobserved variables, then a nonlinear least squares estimator is proposed. 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 established. Lastly, a residual-based empirical process test statistic marked by proper functions of the regressors is proposed for the model checking problem. We further suggest a bootstrap procedure to calculate critical values. Simulation studies demonstrate the performance of the proposed procedure and a real example is analysed to illustrate its practical usage.