Consistent Estimation of the Regression Coefficient Through Weighted Arithmetic Mean of the Inconsistent Estimators in Replicated Ultrastructural Model

The problem of consistent estimation of the slope parameter in an ultrastructural model with replicated observations is considered in this article. A consistent estimator based on a weighted arithmetic mean of two inconsistent least squares estimators is proposed which is independent of any unknown quantity. The efficiency properties of this estimator are studied.     

A Note on the Invariance Law for Surrogate Dimension Reduction

In this article, we give an alternative and much shorter proof for the invariance law of surrogate dimension reduction for measurement error regression that was established by Li and Yin (2007 Li , B. , Yin , X. ( 2007 ). On surrogate dimension reduction for measurement error regression: an invariance law . Ann. Statist. 35 : 2143 – 2172 . [Google Scholar]).     

Objective Priors for Parameters in a Normal Linear Regression with Measurement Error

We investigate certain objective priors for the parameters in a normal linear regression models with one of the explanatory variables subject to measurement error. We first show that the use of the standard non informative prior for normal linear regression without measurement error leads to an improper posterior in the measurement error model. We then derive the Jeffreys prior and reference priors, and show that they lead to proper posteriors. We use simulation study to compare the frequentist performance of the estimates derived using these priors, and the MLE.     

Obtaining a Trapezoidal Distribution

Given a most believed value for a quantity together with upper and lower possible deviations from that value, a rectangular distribution might be used to represent state-of-knowledge about the quantity. If the deviations are themselves known by probability distributions, and the value conditioned on the deviations is rectangular, then the marginal distribution of the value is determined by the distributions of the deviations. Here we show under quite general conditions that conversely, given the marginal distribution, the distributions of the deviations are uniquely determined. The case in which the marginal distribution is trapezoidal is studied in some detail.     

Alternative covariance estimates in a replicated measurement error model with correlated,heteroscedastic errors

The article concerns covariance estimates in a replicated measurement error model with correlated, heteroscedastic errors. Freedman has conjectured that using more of the data will improve estimates of covariance matrices and result in a more efficient estimate of the coefficient of the regression model. The paper confirms the conjecture asymptotically for the case that all random variables are normally distributed, but the gain is not substantial.     

VARIATION REDUCTION AND ROBUST DESIGNS

In this paper we consider two strategies for variation reduction. One of them is the exploitation of interactions. We also discuss the role of experiments in discovering interactions and in particular the use of robust designs to obtain the interaction between control and noise factors. Then we attempt to reduce the variation in a measurement system using a robust design.     

Statistical inference for partially time-varying coefficient errors-in-variables models

This paper studies the partially time-varying coefficient models where some covariates are measured with additive errors. In order to overcome the bias of the usual profile least squares estimation when measurement errors are ignored, we propose a modified profile least squares estimator of the regression parameter and construct estimators of the nonlinear coefficient function and error variance. The proposed three estimators are proved to be asymptotically normal under mild conditions. In addition, we introduce the profile likelihood ratio test and then demonstrate that it follows an asymptotically χ²${\chi }^2$ distribution under the null hypothesis. Finite sample behavior of the estimators is investigated via simulations too.     

A New PC-Based Test for Varian's Weak Separability Conditions

This article develops a new method to evaluate revealed preference separability conditions. In contrast to previous studies, our results generally find weak separability, even when datasets have some measurement error. In addition, revealed preference and weak separability appear robust to measurement error, different price distributions, and alternative preference settings. Measurement error generally results in relatively few violations of revealed preference or weak separability.     

Likelihood-Based Inference for Multivariate Skew-Normal Regression Models

In this article, we present EM algorithms for performing maximum likelihood estimation for three multivariate skew-normal regression models of considerable practical interest. We also consider the restricted estimation of the parameters of certain important special cases of two models. The methodology developed is applied in the analysis of longitudinal data on dental plaque and cholesterol levels.