Measurement error models are studied extensively in the literature. In these models, when the measurement error variance \(\varvec{\Sigma _{\delta \delta }}\) is known, the estimating techniques require positive definiteness of the matrix \(\varvec{S_{xx}-\Sigma _{\delta \delta }}\), 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 \(\varvec{S_{xx}-\Sigma _{\delta \delta }}\) 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.

测量误差模型在文献中得到了广泛的研究。在这些模型中，当测量误差方差\(\varvec{\Sigma _{\delta \delta }}\)已知时，估计技术需要矩阵\(\varvec{S_{xx}-\Sigma _{\delta \delta }}\)，即使这是正定的，如果观察次数很少，它也可能接近奇异。当该矩阵不是正定矩阵时，文献中讨论了替代估计量。本文中，当矩阵\(\varvec{S_{xx}-\Sigma _{\delta \delta }}\)提出了接近奇异值，并且表明这些估计量是一致的，并且具有与早期估计量相同的渐近特性。此外，我们表明我们的估计器在小样本的情况下比早期的估计器工作得更好，并且在大样本的情况下同样好。