Logarithmic calibration for multiplicative distortion measurement errors regression models

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.     

Estimating panel data models in the presence of endogeneity and selection

We consider estimation of panel data models with sample selection when the equation of interest contains endogenous explanatory variables as well as unobserved heterogeneity. Assuming that appropriate instruments are available, we propose several tests for selection bias and two estimation procedures that correct for selection in the presence of endogenous regressors. The tests are based on the fixed effects two-stage least squares estimator, thereby permitting arbitrary correlation between unobserved heterogeneity and explanatory variables. The first correction procedure is parametric and is valid under the assumption that the errors in the selection equation are normally distributed. The second procedure estimates the model parameters semiparametrically using series estimators. In the proposed testing and correction procedures, the error terms may be heterogeneously distributed and serially dependent in both selection and primary equations. Because these methods allow for a rather flexible structure of the error variance and do not impose any nonstandard assumptions on the conditional distributions of explanatory variables, they provide a useful alternative to the existing approaches presented in the literature.     

Connections between Survey Calibration Estimators and Semiparametric Models for Incomplete Data

Survey calibration (or generalized raking) estimators are a standard approach to the use of auxiliary information in survey sampling, improving on the simple Horvitz–Thompson estimator. In this paper we relate the survey calibration estimators to the semiparametric incomplete‐data estimators of Robins and coworkers, and to adjustment for baseline variables in a randomized trial. The development based on calibration estimators explains the “estimated weights” paradox and provides useful heuristics for constructing practical estimators. We present some examples of using calibration to gain precision without making additional modelling assumptions in a variety of regression models.     

Semiparametric econometric estimators for a truncated regression model: a review with an extension

Econometric estimators for a truncated regression model are reviewed. For each estimator, the motivations, the key assumptions, the asymptotic distribution and estimates for the asymptotic variance matrix are presented; also a new estimator is suggested. We select five practical estimators among those, and compare them through a Monte Carlo study where the response variable is simulated but the covariates are drawn from a real data set. Some practical and computational issues are addressed as well.     

Calibration Estimation in Survey Sampling

Calibration estimation, where the sampling weights are adjusted to make certain estimators match known population totals, is commonly used in survey sampling. The generalized regression estimator is an example of a calibration estimator. Given the functional form of the calibration adjustment term, we establish the asymptotic equivalence between the functional-form calibration estimator and an instrumental variable calibration estimator where the instrumental variable is directly determined from the functional form in the calibration equation. Variance estimation based on linearization is discussed and applied to some recently proposed calibration estimators. The results are extended to the estimator that is a solution to the calibrated estimating equation. Results from a limited simulation study are presented.     

Air pollution-mortality models: A demonstration of the effects of random measurement error

Errors of measurement have long been recognized as a chronic problem in statistical analysis. Although there is a vast statistical literature of multiple regression models estimating the air pollution-mortality relationship, this problem has been largely ignored. It is well known that pollution measures contain error, but the consequences of this error for regression estimates is not known. We use Lave and Seskin's air pollution model to demonstrate the consequences of random measurement error. We assume a range of 0% to 50% of the variance of the pollution measures is due to error. We find large differences in the estimated effects on mortality of the pollution variables as well as the other explanatory variables once this measurement error is taken into account. These results cast doubt on the usual regression estimates of the mortality effects of air pollution. More generally our results demonstrate the consequences of random measurement error in the explanatory variable of a multiple regression analysis and the misleading conclusions that may result in policy research if this error is ignored.     

Semiparametric estimation of logistic regression model with missing covariates and outcome

We consider a semiparametric method to estimate logistic regression models with missing both covariates and an outcome variable, and propose two new estimators. The first, which is based solely on the validation set, is an extension of the validation likelihood estimator of Breslow and Cain (Biometrika 75:11–20, 1988). The second is a joint conditional likelihood estimator based on the validation and non-validation data sets. Both estimators are semiparametric as they do not require any model assumptions regarding the missing data mechanism nor the specification of the conditional distribution of the missing covariates given the observed covariates. The asymptotic distribution theory is developed under the assumption that all covariate variables are categorical. The finite-sample properties of the proposed estimators are investigated through simulation studies showing that the joint conditional likelihood estimator is the most efficient. A cable TV survey data set from Taiwan is used to illustrate the practical use of the proposed methodology.     

Model reference adaptive systems applied to regression analyses

The adaptive estimation procedure of model reference adaptive systems is modified and applied to linear models. In general the principle can be used for almost any time series model. Because of the recursive nature of the resulting estimator, it is computationally appealing, especially when a time series is considered as a flow of data. In addition, the estimator turns out to have certain statistical optimality properties.
In the linear regression setting, Ridge estimators turn out to constitute a subclass of the adaptive estimators considered, whereas for unknown measurement variance, the resulting estimators are related to J ames -S tkin type estimators, and have better properties than the latter. The estimator is shown to be strongly consistent and to converge in law to a normal variate under the standard assumptions of linear models. Further it is shown to be admissible and minimax in restricted parameter spaces. The connection between K alman filters and the classical least-squares estimator is also pointed out.     

Likelihood estimation and inference in threshold regression

This paper studies likelihood-based estimation and inference in parametric discontinuous threshold regression models with i.i.d. data. The setup allows heteroskedasticity and threshold effects in both mean and variance. By interpreting the threshold point as a “middle” boundary of the threshold variable, we find that the Bayes estimator is asymptotically efficient among all estimators in the locally asymptotically minimax sense. In particular, the Bayes estimator of the threshold point is asymptotically strictly more efficient than the left-endpoint maximum likelihood estimator and the newly proposed middle-point maximum likelihood estimator. Algorithms are developed to calculate asymptotic distributions and risk for the estimators of the threshold point. The posterior interval is proved to be an asymptotically valid confidence interval and is attractive in both length and coverage in finite samples.     

Mean squared error of ridge estimators in logistic regression

It is well known that the maximum likelihood estimator (MLE) is inadmissible when estimating the multidimensional Gaussian location parameter. We show that the verdict is much more subtle for the binary location parameter. We consider this problem in a regression framework by considering a ridge logistic regression (RR) with three alternative ways of shrinking the estimates of the event probabilities. While it is shown that all three variants reduce the mean squared error (MSE) of the MLE, there is at the same time, for every amount of shrinkage, a true value of the location parameter for which we are overshrinking, thus implying the minimaxity of the MLE in this family of estimators. Little shrinkage also always reduces the MSE of individual predictions for all three RR estimators; however, only the naive estimator that shrinks toward 1/2 retains this property for any generalized MSE (GMSE). In contrast, for the two RR estimators that shrink toward the common mean probability, there is always a GMSE for which even a minute amount of shrinkage increases the error. These theoretical results are illustrated on a numerical example. The estimators are also applied to a real data set, and practical implications of our results are discussed.