In this paper, we prove strong consistency of an estimator by the truncated singular value decomposition for a multivariate errors-in-variables linear regression model with collinearity. This result is an extension of Gleser's proof of the strong consistency of total least squares solutions to the case with modern rank constraints. While the usual discussion of consistency in the absence of solution uniqueness deals with the minimal norm solution, the contribution of this study is to develop a theory that shows the strong consistency of a set of solutions. The proof is based on properties of orthogonal projections, specifically properties of the Rayleigh-Ritz procedure for computing eigenvalues. This makes it suitable for targeting problems where some row vectors of the matrices do not contain noise. Therefore, this paper gives a proof for the regression model with the above condition on the row vectors, resulting in a natural generalization of the strong consistency for the standard TLS estimator.

翻译：本文证明了在多元变量含误差线性回归模型存在共线性时，截断奇异值分解估计量的强相合性。该结果将Gleser关于总体最小二乘解强相合性的证明推广至现代秩约束情形。通常，在解不唯一时的相合性讨论中，研究对象是最小范数解，而本研究的贡献在于建立了一个理论，证明解集的强相合性。证明基于正交投影的性质，具体而言是用于计算特征值的Rayleigh-Ritz过程的性质。这一方法适用于矩阵中某些行向量不含噪声的目标问题。因此，本文针对上述行向量条件下的回归模型给出了证明，从而自然推广了标准TLS估计量的强相合性结果。