Consistent estimation in measurement error models with near singular covariance

Measurement error models are studied extensively in the literature. In these models, when the measurement error variance Σδδ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma _{\delta \delta }}$$\end{document} is known, the estimating techniques require positive definiteness of the matrix Sxx-Σδδ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{S_{xx}-\Sigma _{\delta \delta }}$$\end{document}, even when this is positive definite, it might be near singular if the number of observations is small. There are alternative estimators discussed in literature when this matrix is not positive definite. In this paper, estimators when the matrix Sxx-Σδδ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{S_{xx}-\Sigma _{\delta \delta }}$$\end{document} is near singular are proposed and it is shown that these estimators are consistent and have the same asymptotic properties as the earlier ones. In addition, we show that our estimators work far better than the earlier estimators in case of small samples and equally good for large samples.